Question 11611: 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=[-2, -2, 0, 3, 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. operator [a,b]=[2, -2] means a*u_x+b*u_y; source=-2, x=-2, y=0. 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=[-2, -2, 0, 3, 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. operator [a,b]=[2, -2] means a*u_x+b*u_y; source=-2, x=-2, y=0. 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=[-2, -2, 0, 3, 0, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-2)+(-2)*x+(0)*y+(3)*x^2+(0)*x*y+(-2)*y^2

PDE x slice fixed coordinate squared := mul(0, 0)

PDE x slice fixed coordinate squared = 0

PDE x slice fixed linear term := mul(0, 0)

PDE x slice fixed linear term = 0

PDE x slice fixed square term := mul(-2, 0)

PDE x slice fixed square term = 0

PDE x slice constant coefficient := sum(-2, 0, 0)

PDE x slice constant coefficient = -2

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

PDE x slice cross contribution = 0

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

PDE x slice linear coefficient = -2

PDE x slice:vary x;fix y=0;original coefficients=[-2, -2, 0, 3, 0, -2]

ascending x slice coefficients=[-2, -2, 3]

PDE x partial:g(t)=(-2)+(-2)*t+(3)*t^2;point=-2;h!=0

g(-2+h)-g(-2)=(-2)*h+(3)*(2*(-2)*h+h^2)

PDE x partial twice quadratic coefficient := mul(2, 3)

PDE x partial twice quadratic coefficient = 6

PDE x partial derivative point term := mul(6, -2)

PDE x partial derivative point term = -12

PDE x partial derivative value := sum(-2, -12)

PDE x partial derivative value = -14

PDE x partial:h!=0;increment=h*((-2)+2*(3)*(-2)+(3)*h)

difference quotient=-14+(3)*h

PDE x partial:quotient=-14+(3)*h;epsilon>0;delta=epsilon/3

0<abs(h)<delta gives abs(quotient-(-14))=abs(3)*abs(h)<epsilon;limit=-14

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

PDE y slice fixed coordinate squared = 4

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

PDE y slice fixed linear term = 4

PDE y slice fixed square term := mul(3, 4)

PDE y slice fixed square term = 12

PDE y slice constant coefficient := sum(-2, 4, 12)

PDE y slice constant coefficient = 14

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

PDE y slice cross contribution = 0

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

PDE y slice linear coefficient = 0

PDE y slice:vary y;fix x=-2;original coefficients=[-2, -2, 0, 3, 0, -2]

ascending y slice coefficients=[14, 0, -2]

PDE y partial:g(t)=(14)+(0)*t+(-2)*t^2;point=0;h!=0

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

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

PDE y partial twice quadratic coefficient = -4

PDE y partial derivative point term := mul(-4, 0)

PDE y partial derivative point term = 0

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

PDE y partial derivative value = 0

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

difference quotient=0+(-2)*h

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

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

first order term0 := mul(2, -14)

first order term0 = -28

first order term1 := mul(-2, 0)

first order term1 = 0

first order operator sum := sum(-28, 0)

first order operator sum = -28

first order signed residual := sub(-28, -2)

first order signed residual = -26

query point=[-2, 0];operator value=-28;source=-2;signed residual=-26

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

([-2, -2, 0, 3, 0, -2], [2, -2], -2, 0, -14, 0, -2, -26, 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.