Question 11612: One-Point Residual of a Declared First-Order PDE Operator

Original question and givens

Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 1, 4, 1, -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]=[3, -1] means a*u_x+b*u_y; source=-1, x=-1, y=1. 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=[-1, -1, 1, 4, 1, -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]=[3, -1] means a*u_x+b*u_y; source=-1, x=-1, y=1. 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

Step-by-step derivation

original coefficients=[-1, -1, 1, 4, 1, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-1)+(-1)*x+(1)*y+(4)*x^2+(1)*x*y+(-1)*y^2

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

PDE x slice fixed coordinate squared = 1

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

PDE x slice fixed linear term = 1

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

PDE x slice fixed square term = -1

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

PDE x slice constant coefficient = -1

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

PDE x slice cross contribution = 1

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

PDE x slice linear coefficient = 0

PDE x slice:vary x;fix y=1;original coefficients=[-1, -1, 1, 4, 1, -1]

ascending x slice coefficients=[-1, 0, 4]

PDE x partial:g(t)=(-1)+(0)*t+(4)*t^2;point=-1;h!=0

g(-1+h)-g(-1)=(0)*h+(4)*(2*(-1)*h+h^2)

PDE x partial twice quadratic coefficient := mul(2, 4)

PDE x partial twice quadratic coefficient = 8

PDE x partial derivative point term := mul(8, -1)

PDE x partial derivative point term = -8

PDE x partial derivative value := sum(0, -8)

PDE x partial derivative value = -8

PDE x partial:h!=0;increment=h*((0)+2*(4)*(-1)+(4)*h)

difference quotient=-8+(4)*h

PDE x partial:quotient=-8+(4)*h;epsilon>0;delta=epsilon/4

0<abs(h)<delta gives abs(quotient-(-8))=abs(4)*abs(h)<epsilon;limit=-8

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

PDE y slice fixed coordinate squared = 1

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

PDE y slice fixed linear term = 1

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

PDE y slice fixed square term = 4

PDE y slice constant coefficient := sum(-1, 1, 4)

PDE y slice constant coefficient = 4

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

PDE y slice cross contribution = -1

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

PDE y slice linear coefficient = 0

PDE y slice:vary y;fix x=-1;original coefficients=[-1, -1, 1, 4, 1, -1]

ascending y slice coefficients=[4, 0, -1]

PDE y partial:g(t)=(4)+(0)*t+(-1)*t^2;point=1;h!=0

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

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

PDE y partial twice quadratic coefficient = -2

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

PDE y partial derivative point term = -2

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

PDE y partial derivative value = -2

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

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

PDE 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

first order term0 := mul(3, -8)

first order term0 = -24

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

first order term1 = 2

first order operator sum := sum(-24, 2)

first order operator sum = -22

first order signed residual := sub(-22, -1)

first order signed residual = -21

query point=[-1, 1];operator value=-22;source=-1;signed residual=-21

satisfied at declared point=No;no general solution claim

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

([-1, -1, 1, 4, 1, -1], [3, -1], -1, 1, -8, -2, -1, -21, 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.