Calculus
Lab: partials, gradient and tangent plane
Connect trace slopes with a surface’s linear model. Rotate direction, reduce h and compare exact derivatives, secants and error. Two counterexamples show why partial derivatives alone are insufficient.
Derivatives and differentiability
One halving per second, down to h=10⁻⁴. A limit requires h→0 from both sides; one small value is not a proof.
Surface, trace tangents and plane
Green surface; orange plane and directional tangent; blue x-tangent and purple y-tangent. In xy: purple gradient, orange u, sampled green level. The gradient arrow is normalized for display; its norm is listed numerically.
Level through P and vectors in the xy-plane
Trace in direction u
Increment, secants and error
One of P±hu is outside the domain: its quotient cannot be computed. Reduce h.
Value table
Derivatives and case classifications are analytical; finite differences, surfaces and levels use floating-point samples. Relative error on one direction does not prove differentiability. A central difference can hide failure of a directional derivative. Window and vertical scale do not change the domain; high or invalid faces are omitted, not flattened.