# Question 11602: One-Point Residual of a Declared First-Order PDE Operator

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0022-11602 · q-20261013-advance-20261024-r06-partial-differential-equation-0002



## Original question and givens

Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-11, -1, -2, 2, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, -1] means a*u_x+b*u_y; source=-11, x=-1, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).


- G001 — Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-11, -1, -2, 2, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, -1] means a*u_x+b*u_y; source=-11, x=-1, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).





## 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-differential-equation](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/)





## Step-by-step derivation

S001 · impliesBefore

original coefficients=[-11, -1, -2, 2, 0, -1];basis=[1,x,y,x^2,xy,y^2]

After

u(x,y)=(-11)+(-1)*x+(-2)*y+(2)*x^2+(0)*x*y+(-1)*y^2

Conditions: none

Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic.

Step identifiers and dependenciesRule: bivariate_scalar_polynomial_basis_roles · Uses: G001





S002 · calculationBefore

PDE x slice fixed coordinate squared := mul(-2, -2)

After

PDE x slice fixed coordinate squared = 4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

PDE x slice fixed linear term := mul(-2, -2)

After

PDE x slice fixed linear term = 4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

PDE x slice fixed square term := mul(-1, 4)

After

PDE x slice fixed square term = -4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

PDE x slice constant coefficient := sum(-11, 4, -4)

After

PDE x slice constant coefficient = -11

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

PDE x slice cross contribution := mul(0, -2)

After

PDE x slice cross contribution = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

PDE x slice linear coefficient := sum(-1, 0)

After

PDE x slice linear coefficient = -1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

PDE x slice:vary x;fix y=-2;original coefficients=[-11, -1, -2, 2, 0, -1]

After

ascending x slice coefficients=[-11, -1, 2]

Conditions: none

Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.

Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S007





S009 · impliesBefore

PDE x partial:g(t)=(-11)+(-1)*t+(2)*t^2;point=-1;h!=0

After

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

Conditions: none

Cancel constants and expand(z+h)^2−z^2=2zh+h^2.Do not substitute h=0 at this stage.

Step identifiers and dependenciesRule: slice_increment_exact_expansion · Uses: G001, S008





S010 · calculationBefore

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

After

PDE x partial twice quadratic coefficient = 4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

PDE x partial derivative point term := mul(4, -1)

After

PDE x partial derivative point term = -4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

PDE x partial derivative value := sum(-1, -4)

After

PDE x partial derivative value = -5

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

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

After

difference quotient=-5+(2)*h

Conditions: none

Factor out the whole h and cancel only under h≠0.The quotient still has D*h;a finite difference is not automatically the derivative.

Step identifiers and dependenciesRule: cancel_only_nonzero_increment · Uses: G001, S012





S014 · impliesBefore

PDE x partial:quotient=-5+(2)*h;epsilon>0;delta=epsilon/2

After

0<abs(h)<delta gives abs(quotient-(-5))=abs(2)*abs(h)<epsilon;limit=-5

Conditions: none

If D=0 the error is zero and any positive delta works.Otherwise choose epsilon/|D|.This covers every sufficiently small nonzero h of either sign,not sampled increments.

Step identifiers and dependenciesRule: slice_two_sided_epsilon_delta_limit · Uses: G001, S013





S015 · calculationBefore

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

After

PDE y slice fixed coordinate squared = 1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

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

After

PDE y slice fixed linear term = 1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

PDE y slice fixed square term := mul(2, 1)

After

PDE y slice fixed square term = 2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

PDE y slice constant coefficient := sum(-11, 1, 2)

After

PDE y slice constant coefficient = -8

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

PDE y slice cross contribution := mul(0, -1)

After

PDE y slice cross contribution = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

PDE y slice linear coefficient := sum(-2, 0)

After

PDE y slice linear coefficient = -2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · impliesBefore

PDE y slice:vary y;fix x=-1;original coefficients=[-11, -1, -2, 2, 0, -1]

After

ascending y slice coefficients=[-8, -2, -1]

Conditions: none

Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.

Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S020





S022 · impliesBefore

PDE y partial:g(t)=(-8)+(-2)*t+(-1)*t^2;point=-2;h!=0

After

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

Conditions: none

Cancel constants and expand(z+h)^2−z^2=2zh+h^2.Do not substitute h=0 at this stage.

Step identifiers and dependenciesRule: slice_increment_exact_expansion · Uses: G001, S021





S023 · calculationBefore

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

After

PDE y partial twice quadratic coefficient = -2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

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

After

PDE y partial derivative point term = 4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

PDE y partial derivative value := sum(-2, 4)

After

PDE y partial derivative value = 2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

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

After

difference quotient=2+(-1)*h

Conditions: none

Factor out the whole h and cancel only under h≠0.The quotient still has D*h;a finite difference is not automatically the derivative.

Step identifiers and dependenciesRule: cancel_only_nonzero_increment · Uses: G001, S025





S027 · impliesBefore

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

After

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

Conditions: none

If D=0 the error is zero and any positive delta works.Otherwise choose epsilon/|D|.This covers every sufficiently small nonzero h of either sign,not sampled increments.

Step identifiers and dependenciesRule: slice_two_sided_epsilon_delta_limit · Uses: G001, S026





S028 · calculationBefore

first order term0 := mul(2, -5)

After

first order term0 = -10

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

first order term1 := mul(-1, 2)

After

first order term1 = -2

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

first order operator sum := sum(-10, -2)

After

first order operator sum = -12

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

first order signed residual := sub(-12, -11)

After

first order signed residual = -1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

query point=[-1, -2];operator value=-12;source=-11;signed residual=-1

After

satisfied at declared point=No;no general solution claim

Conditions: none

This flag tests the residual at the declared point.A point pass does not generally prove a global solution or existence,uniqueness,regularity or well-posedness.A global identity requires a separate full symbolic residual audit.

Step identifiers and dependenciesRule: PDE_point_obligation_not_global_solution · Uses: G001, S031









## Reverse checks


- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S031 · premise_and_result_audit — Recompute first order signed residual with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Recompute first order operator sum with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Recompute first order term1 with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute first order term0 with exact rational arithmetic, not rounded decimals.
- 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 PDE y partial derivative value with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute PDE y partial derivative point term with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Recompute PDE 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 PDE y slice linear coefficient with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute PDE y slice cross contribution with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Recompute PDE y slice constant coefficient with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute PDE y slice fixed square term with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute PDE y slice fixed linear term with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute PDE y 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 PDE x partial derivative value with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute PDE x partial derivative point term with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute PDE 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 PDE x slice linear coefficient with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute PDE x slice cross contribution with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute PDE x slice constant coefficient with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute PDE x slice fixed square term with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute PDE x slice fixed linear term with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute PDE x 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

([-11, -1, -2, 2, 0, -1], [2, -1], -1, -2, -5, 2, -11, -1, No)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 698e0129612ae018dcd2fcc0268e18b782804d6d3d211838278a93601bc64043
- candidate record SHA-256: b8777e376fa5c49c6bf52959adb115317289fb67db2fe69e2728450483649c6a
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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