Question 11512: Collect the Two Coordinate Partials in Their Declared Order

Original question and givens

Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

• G001 — Use coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. x=-1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-partial-derivative

Step-by-step derivation

original coefficients=[-1, -1, 1, 1, 1, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-1)+(-1)*x+(1)*y+(1)*x^2+(1)*x*y+(-1)*y^2

x partial slice fixed coordinate squared := mul(1, 1)

x partial slice fixed coordinate squared = 1

x partial slice fixed linear term := mul(1, 1)

x partial slice fixed linear term = 1

x partial slice fixed square term := mul(-1, 1)

x partial slice fixed square term = -1

x partial slice constant coefficient := sum(-1, 1, -1)

x partial slice constant coefficient = -1

x partial slice cross contribution := mul(1, 1)

x partial slice cross contribution = 1

x partial slice linear coefficient := sum(-1, 1)

x partial slice linear coefficient = 0

x partial slice:vary x;fix y=1;original coefficients=[-1, -1, 1, 1, 1, -1]

ascending x slice coefficients=[-1, 0, 1]

x partial:g(t)=(-1)+(0)*t+(1)*t^2;point=-1;h!=0

g(-1+h)-g(-1)=(0)*h+(1)*(2*(-1)*h+h^2)

x partial twice quadratic coefficient := mul(2, 1)

x partial twice quadratic coefficient = 2

x partial derivative point term := mul(2, -1)

x partial derivative point term = -2

x partial derivative value := sum(0, -2)

x partial derivative value = -2

x partial:h!=0;increment=h*((0)+2*(1)*(-1)+(1)*h)

difference quotient=-2+(1)*h

x partial:quotient=-2+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(-2))=abs(1)*abs(h)<epsilon;limit=-2

y partial slice fixed coordinate squared := mul(-1, -1)

y partial slice fixed coordinate squared = 1

y partial slice fixed linear term := mul(-1, -1)

y partial slice fixed linear term = 1

y partial slice fixed square term := mul(1, 1)

y partial slice fixed square term = 1

y partial slice constant coefficient := sum(-1, 1, 1)

y partial slice constant coefficient = 1

y partial slice cross contribution := mul(1, -1)

y partial slice cross contribution = -1

y partial slice linear coefficient := sum(1, -1)

y partial slice linear coefficient = 0

y partial slice:vary y;fix x=-1;original coefficients=[-1, -1, 1, 1, 1, -1]

ascending y slice coefficients=[1, 0, -1]

y partial:g(t)=(1)+(0)*t+(-1)*t^2;point=1;h!=0

g(1+h)-g(1)=(0)*h+(-1)*(2*(1)*h+h^2)

y partial twice quadratic coefficient := mul(2, -1)

y partial twice quadratic coefficient = -2

y partial derivative point term := mul(-2, 1)

y partial derivative point term = -2

y partial derivative value := sum(0, -2)

y partial derivative value = -2

y partial:h!=0;increment=h*((0)+2*(-1)*(1)+(-1)*h)

difference quotient=-2+(-1)*h

y partial:quotient=-2+(-1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(-2))=abs(-1)*abs(h)<epsilon;limit=-2

x partial=-2;y partial=-2

gradient in declared x,y order=[-2, -2]

Reverse checks

• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute y partial derivative value with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute y partial derivative point term with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute y partial twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute y partial slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute y partial slice cross contribution with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute y partial slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute y partial slice fixed square term with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute y partial slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute y partial slice fixed coordinate squared with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute x partial derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute x partial derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute x partial twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute x partial slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute x partial slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute x partial slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute x partial slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute x partial slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute x partial slice fixed coordinate squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.