# Question 11680: Second-Order Residual with an Explicit Mixed-Term Convention

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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.

R0022-11680 · q-20261013-advance-20261024-r06-partial-differential-equation-0030



## Original question and givens

Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-8, 2, 1, 1, 0, 2] 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. principal [a,b,c]=[2, 2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-8, x=2, y=1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).


- G001 — Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-8, 2, 1, 1, 0, 2] 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. principal [a,b,c]=[2, 2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-8, x=2, y=1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the 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=[-8, 2, 1, 1, 0, 2];basis=[1,x,y,x^2,xy,y^2]

After

u(x,y)=(-8)+(2)*x+(1)*y+(1)*x^2+(0)*x*y+(2)*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

u_xx constant := mul(2, 1)

After

u_xx constant = 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, S001





S003 · calculationBefore

u_yy constant := mul(2, 2)

After

u_yy constant = 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 · impliesBefore

u coefficients=[-8, 2, 1, 1, 0, 2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

After

u_xx=2;u_xy=0;u_yy=4

Conditions: none

Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem.

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





S005 · impliesBefore

principal=[2, 2, 1];point=[2, 1]

After

operator=a*u_xx+b*u_xy+c*u_yy;not 2*b*u_xy

Conditions: none

The mixed term is b*u_xy,not the alternative convention2b*u_xy.Retain the query point roles even though these second derivatives are constant for this polynomial.

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





S006 · calculationBefore

second order term0 := mul(2, 2)

After

second order term0 = 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, S005





S007 · calculationBefore

second order term1 := mul(2, 0)

After

second order term1 = 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, S006





S008 · calculationBefore

second order term2 := mul(1, 4)

After

second order term2 = 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, S007





S009 · calculationBefore

second order operator sum := sum(4, 0, 4)

After

second order operator sum = 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, S008





S010 · calculationBefore

second order signed residual := sub(8, -8)

After

second order signed residual = 16

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 · impliesBefore

query point=[2, 1];operator value=8;source=-8;signed residual=16

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, S010









## Reverse checks


- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Recompute second order signed residual with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute second order operator sum with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute second order term2 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute second order term1 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute second order term0 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute u_yy constant with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute u_xx constant 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

([-8, 2, 1, 1, 0, 2], [2, 2, 1], 2, 1, 2, 0, 4, -8, 16, 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: 4e4b506d0b7aee41426f0d4ff6fd6be2cf8be7aa860e8c70094309716012c414
- candidate record SHA-256: 74aa1d1a6a8c448cdf8b4d2c6ce86df839a85eef44257c4a2cfcfd2967cd4d6e
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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