Coordinate gradients, stationarity and differentiability

Goal and scope

Compute the gradient of a specified bivariate polynomial and audit two false inferences.

Fixed standard Euclidean coordinate order (x,y), with no implicit optimization or descent certificate.

Givens and symbols

• algebra
• polynomial
• definition_here

• G1 — f(x,y)=x^2+x*y+2*y^2-4*x+2*y, coordinates ordered (x,y), evaluation point p=(1,-1).

• \nabla f — gradient in fixed coordinate order
• p — exact evaluation point

Rules and sources

• mko_definition · cited — Source semantic paraphrase: the gradient contains each coordinate partial in fixed order; existing partials and stationarity do not automatically imply differentiability or an extremum. · mko-gradient-vector
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.
• definition_here · prerequisite — In fixed standard Euclidean coordinates, the gradient is the ordered vector of all coordinate partials; a stationary point has all gradient coordinates zero.

Main derivation

f=x^2+xy+2y^2-4x+2y

\partial_x f=2x+y-4

f=x^2+xy+2y^2-4x+2y

\partial_y f=x+4y+2

\nabla f=(\partial_x f,\partial_y f)

\nabla f(x,y)=(2x+y-4,x+4y+2)

p=(1,-1)

\nabla f(p)=(-3,-1)\ne(0,0)

2x+y=4,\quad x+4y=-2

(x,y)=(18/7,-8/7),\quad\nabla f=0

s(x,y)=x^2-y^2

\nabla s(0,0)=0,\quad s(t,0)>0>s(0,t)\ (t\ne0)

h(x,y)=x^3/(x^2+y^2)\ ((x,y)\ne0),\quad h(0,0)=0

\partial_xh(0,0)=1,\quad\partial_yh(0,0)=0

h(t,t)=t/2,\quad Dh_{\text{candidate}}(t,t)=t

\frac{|h(t,t)-t|}{\|(t,t)\|_2}=\frac1{2\sqrt2}\not\to0

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S07 — The necessary differentiability remainder condition explicitly refutes existing partials implying differentiability.
• S07 · premise_audit · G1 — Coordinate partials exist, but a full-direction linear remainder is not yet proved.
• S06 · premise_audit · G1 — Both directional values are supplied instead of only a saddle label.
• S05 · inverse_equivalence · S03 — This proves stationarity only, without declaring a minimum merely from zero gradient.
• S04 · premise_audit · G1, S03 — The nonzero gradient is not converted into an unspecified descent step or optimum classification.
• S03 · premise_audit · S01, S02 — One partial alone cannot be called the complete gradient.
• S02 · premise_audit · G1 — 4y comes from 2y^2 without dropping a factor two.
• S01 · premise_audit · G1 — The x-derivatives of 2y^2 and 2y vanish.

Error controls and negative controls

Coordinate order

Both symbolic differentiation and evaluation use (x,y).

Separate obligations

Keep stationarity, differentiability, extrema and global optimality separate.

• A gradient needs only one coordinate. → reject — An R^2 source requires both ordered partials.
• A zero gradient guarantees a minimum. → reject — The saddle x^2-y^2 has both signs near the origin.
• All partials existing imply differentiability. → reject — The diagonal remainder at S07-S08 does not vanish.

Result and limitations

The gradient is (2x+y-4,x+4y+2), at p it is (-3,-1); zero gradient and existing partials do not imply extrema and differentiability respectively.

Verification and source binding

4 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 5d7de98af659977808b6bdfbc360956ccd0ca6a6d166fe2b4f428078f3673915
• original source SHA-256: 5d7de98af659977808b6bdfbc360956ccd0ca6a6d166fe2b4f428078f3673915
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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