Gradient vector

Mathematical statement

For f:R^n→R, the gradient is (partial_1 f,...,partial_n f), with coordinate order part of the data. Exact rational polynomial terms [coefficient,exponent vector] give symbolic component polynomials and exact rational-point values. Zero gradient certifies stationarity, not necessarily a local/global extremum. Existence of all coordinate partials does not generally imply differentiability. Boundaries include the norm at the origin, a saddle x^2−y^2, an incomplete coordinate list and scale*x^3/(x^2+y^2) extended by0, whose coordinate partials exist but whose diagonal remainder refutes differentiability.

Representative formula (exact TeX)

G(f,x)=v

Assumptions

• Dimension, exponent vectors and point arity match exactly; combine equal monomials canonically first.
• The basis/metric is standard Euclidean, with no silent coordinate or metric change.

Symbol scope

• G — gradient operator
• f — multivariable real function
• x — coordinate point
• v — coordinate gradient vector

Intuition, definition, and misconceptions

The symbolic gradient retains dimension and coordinate order.

Point evaluation uses exact rationals.

Stationarity, differentiability and extremality are different obligations.

Do not infer differentiability merely from existing partials, or a minimum/global optimum from a zero gradient.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Partial derivative — Components are coordinate derivatives.
• Plane vectors in a fixed basis — Component order forms a vector.
• Multivariable real-valued function — The domain is multivariable and codomain scalar.

Verification boundary

This slice supplies no gradient-descent convergence, Lagrange-multiplier or Hessian classification automatically.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add gradient descent and Lagrange-multiplier necessary conditions.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.