Multivariable real-valued function

Mathematical statement

This entry initially restricts attention to explicit slices of bivariate real polynomials f:R²→R, with coefficients ordered as [1,x,y,x²,x*y,y²]. Function qualification requires a declared domain and coordinate roles; continuity, differentiability and vector-valued codomains are not automatically asserted.

Representative formula (exact TeX)

f(x,y)

Assumptions

• Exercises use six integer coefficients and integer coordinate points.
• Rectangle-domain audits check only an explicitly declared closed rectangle. Although polynomial expressions can be globally defined, the exercise still follows its declared domain.

Symbol scope

• f — bivariate real-valued function
• x — first coordinate
• y — second coordinate

Intuition, definition, and misconceptions

A fixed coefficient order is necessary for exact recomputation.

Rectangle-domain membership requires checking the x and y bounds separately.

Fixing y gives only one x-slice; a finite slice table does not determine the full function.

Do not swap x and y coefficients or use a few point values to establish global function equivalence, continuity or differentiability.

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

• Function mapping — Each admissible input receives exactly one real output.
• Plane vectors in a fixed basis — A multivariable input can be viewed as a real coordinate vector.
• Coordinate plane — Bivariate inputs use ordered coordinates.

Verification boundary

Only R²→R polynomial slices are checked; partial derivatives and general regularity are not established.

not_formalized

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

Next obligation

Add partial derivatives, gradients and PDEs.

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.