Gradient

The central idea

For differentiable $f:\mathbb R^n\to\mathbb R$, the gradient is $\nabla f=\langle f_{x_1},\ldots,f_{x_n}\rangle\in\mathbb R^n$. It represents the derivative as a vector via $Df(\mathbf a)[\mathbf h]=\nabla f(\mathbf a)\cdot\mathbf h$. At a regular level set, the gradient is normal to tangent directions.

Definitions, hypotheses, and notation

A gradient depends on the chosen inner product and coordinates; in ordinary Cartesian Euclidean space, the standard dot product identifies the derivative covector with this familiar vector. The chain rule along a curve, $d(f\circ\mathbf r)/dt=\nabla f\cdot\mathbf r'$, proves normality to level sets because $f$ is constant along a level curve.

At points where the gradient vanishes, every first-order directional derivative is zero, so second-order or constraint analysis is needed. At a singular level set, normal conclusions may fail because there is no nonzero gradient to determine a unique tangent hyperplane.

Conceptual meaning

The gradient points in the direction of steepest increase, with magnitude equal to the maximum rate per unit distance. Its negative points toward steepest decrease. On contour maps it crosses level curves orthogonally toward larger labels.

A dependable method and decision rule

1. Compute each partial derivative in coordinate order.
2. Evaluate all components at the same point.
3. Treat the result as a vector in the input space.
4. Normalize only when a unit direction is requested.
5. Use dot products for directional rates or normals for level-set tangents.

Fully worked example

Common mistakes and why they fail