Ordered Euclidean Derivatives, Exact Values and Mode-Specific Counterexamples
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batch-20261015-advance-20261113-r06-gradient-vector-foundations · 2026-10-15 · 100 questions
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1. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[1, [2, 1]], [2, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
2. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
3. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
4. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[4, [2, 1, 0]], [5, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
5. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[5, [2, 1]], [6, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
6. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[6, [2, 1, 0]], [7, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
7. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[7, [2, 1]], [8, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
8. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
9. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[9, [2, 1]], [10, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
10. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[10, [2, 1, 0]], [11, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
11. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[11, [2, 1]], [12, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
12. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[12, [2, 1, 0]], [13, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
13. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[13, [2, 1]], [14, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
14. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[14, [2, 1, 0]], [15, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
15. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
16. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[16, [2, 1, 0]], [17, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
17. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[17, [2, 1]], [18, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
18. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
19. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[19, [2, 1]], [20, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
20. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[20, [2, 1, 0]], [21, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
21. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[21, [2, 1]], [22, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
22. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[22, [2, 1, 0]], [23, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
23. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[23, [2, 1]], [24, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
24. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=3, raw polynomial terms=[[24, [2, 1, 0]], [25, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
25. The Symbolic Gradient Keeps Every Euclidean Coordinate in Order
Use dimension=2, raw polynomial terms=[[25, [2, 1]], [26, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).
View answer and explanation
(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)
Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-symbolic-polynomial · candidate
26. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[1, [2, 1]], [2, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[1/1, 2/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
27. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[2/1, 3/2, 4/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
28. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[3/1, 4/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
29. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[4, [2, 1, 0]], [5, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[4/1, 5/2, 6/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
30. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[5, [2, 1]], [6, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[5/1, 6/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
31. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[6, [2, 1, 0]], [7, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[6/1, 7/2, 8/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
32. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[7, [2, 1]], [8, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[7/1, 8/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
33. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[8/1, 9/2, 10/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
34. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[9, [2, 1]], [10, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[9/1, 10/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
35. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[10, [2, 1, 0]], [11, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[10/1, 11/2, 12/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
36. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[11, [2, 1]], [12, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[11/1, 12/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
37. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[12, [2, 1, 0]], [13, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[12/1, 13/2, 14/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
38. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[13, [2, 1]], [14, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[13/1, 14/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
39. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[14, [2, 1, 0]], [15, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[14/1, 15/2, 16/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
40. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[15/1, 16/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
41. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[16, [2, 1, 0]], [17, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[16/1, 17/2, 18/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
42. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[17, [2, 1]], [18, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[17/1, 18/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
43. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[18/1, 19/2, 20/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
44. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[19, [2, 1]], [20, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[19/1, 20/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
45. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[20, [2, 1, 0]], [21, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[20/1, 21/2, 22/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
46. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[21, [2, 1]], [22, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[21/1, 22/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
47. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[22, [2, 1, 0]], [23, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[22/1, 23/2, 24/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
48. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[23, [2, 1]], [24, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[23/1, 24/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
49. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=3, raw polynomial terms=[[24, [2, 1, 0]], [25, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[24/1, 25/2, 26/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
50. Evaluate Every Coordinate Derivative at the Same Exact Rational Point
Use dimension=2, raw polynomial terms=[[25, [2, 1]], [26, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[25/1, 26/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).
View answer and explanation
(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)
Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-rational-point · candidate
51. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[1, [2, 0]], [2, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
52. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[2, [2, 0]], [3, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
53. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[3, [2, 0]], [4, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
54. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[4, [2, 0]], [5, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
55. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[5, [2, 0]], [6, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
56. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[6, [2, 0]], [7, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
57. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[7, [2, 0]], [8, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
58. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[8, [2, 0]], [9, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
59. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[9, [2, 0]], [10, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
60. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[10, [2, 0]], [11, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
61. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[11, [2, 0]], [12, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
62. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[12, [2, 0]], [13, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
63. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[13, [2, 0]], [14, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
64. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[14, [2, 0]], [15, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
65. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[15, [2, 0]], [16, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
66. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[16, [2, 0]], [17, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
67. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[17, [2, 0]], [18, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
68. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[18, [2, 0]], [19, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
69. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[19, [2, 0]], [20, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
70. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[20, [2, 0]], [21, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
71. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[21, [2, 0]], [22, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
72. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[22, [2, 0]], [23, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
73. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[23, [2, 0]], [24, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
74. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[24, [2, 0]], [25, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
75. A Zero Gradient Certifies Stationarity, Not an Automatic Extremum
Use dimension=2, raw polynomial terms=[[25, [2, 0]], [26, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).
The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function.
View answer and explanation
(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)
Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-stationary-audit · candidate
76. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=1, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
77. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=2, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
78. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=3, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], Yes, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], Yes, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
79. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=4, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
80. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=5, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
81. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=6, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
82. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=7, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], Yes, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], Yes, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
83. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=8, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
84. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=9, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
85. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=10, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
86. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=11, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
87. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=12, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
88. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=13, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
89. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=14, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
90. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=15, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], Yes, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], Yes, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
91. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=16, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
92. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=17, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
93. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=18, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
94. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=19, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], Yes, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], Yes, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
95. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=20, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
96. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=21, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
97. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=nondifferentiable_norm, positive scale=22, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
98. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=partials_exist_not_differentiable, positive scale=23, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], Yes, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], Yes, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
99. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=saddle_stationary, positive scale=24, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
100. Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts
Use mode=incomplete_coordinate_list, positive scale=25, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).
mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used.
View answer and explanation
(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)
incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3).
Reviewed step-by-step derivation
Gradient vector · qf-mod44-gradient-boundary-counterexample · candidate
Canonical question JSONL · English JSONL · Validation report