Question 11607: 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=[-6, -1, 3, 3, -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]=[1, -1] means a*u_x+b*u_y; source=-6, x=-1, y=3. 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=[-6, -1, 3, 3, -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]=[1, -1] means a*u_x+b*u_y; source=-6, x=-1, y=3. 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=[-6, -1, 3, 3, -1, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-6)+(-1)*x+(3)*y+(3)*x^2+(-1)*x*y+(-1)*y^2

PDE x slice fixed coordinate squared := mul(3, 3)

PDE x slice fixed coordinate squared = 9

PDE x slice fixed linear term := mul(3, 3)

PDE x slice fixed linear term = 9

PDE x slice fixed square term := mul(-1, 9)

PDE x slice fixed square term = -9

PDE x slice constant coefficient := sum(-6, 9, -9)

PDE x slice constant coefficient = -6

PDE x slice cross contribution := mul(-1, 3)

PDE x slice cross contribution = -3

PDE x slice linear coefficient := sum(-1, -3)

PDE x slice linear coefficient = -4

PDE x slice:vary x;fix y=3;original coefficients=[-6, -1, 3, 3, -1, -1]

ascending x slice coefficients=[-6, -4, 3]

PDE x partial:g(t)=(-6)+(-4)*t+(3)*t^2;point=-1;h!=0

g(-1+h)-g(-1)=(-4)*h+(3)*(2*(-1)*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, -1)

PDE x partial derivative point term = -6

PDE x partial derivative value := sum(-4, -6)

PDE x partial derivative value = -10

PDE x partial:h!=0;increment=h*((-4)+2*(3)*(-1)+(3)*h)

difference quotient=-10+(3)*h

PDE x partial:quotient=-10+(3)*h;epsilon>0;delta=epsilon/3

0<abs(h)<delta gives abs(quotient-(-10))=abs(3)*abs(h)<epsilon;limit=-10

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(3, 1)

PDE y slice fixed square term = 3

PDE y slice constant coefficient := sum(-6, 1, 3)

PDE y slice constant coefficient = -2

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

PDE y slice cross contribution = 1

PDE y slice linear coefficient := sum(3, 1)

PDE y slice linear coefficient = 4

PDE y slice:vary y;fix x=-1;original coefficients=[-6, -1, 3, 3, -1, -1]

ascending y slice coefficients=[-2, 4, -1]

PDE y partial:g(t)=(-2)+(4)*t+(-1)*t^2;point=3;h!=0

g(3+h)-g(3)=(4)*h+(-1)*(2*(3)*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, 3)

PDE y partial derivative point term = -6

PDE y partial derivative value := sum(4, -6)

PDE y partial derivative value = -2

PDE y partial:h!=0;increment=h*((4)+2*(-1)*(3)+(-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(1, -10)

first order term0 = -10

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

first order term1 = 2

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

first order operator sum = -8

first order signed residual := sub(-8, -6)

first order signed residual = -2

query point=[-1, 3];operator value=-8;source=-6;signed residual=-2

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

([-6, -1, 3, 3, -1, -1], [1, -1], -1, 3, -10, -2, -6, -2, 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.