# Difference between revisions of "Gradient" This article/section deals with mathematical concepts appropriate for late high school or early college.

In mathematics, the gradient is a vector associated to a point of a differentiable function which takes real values. Specifically, the gradient at is a vector in which points in the direction in which increases most rapidly at . The magnitude of the gradient at is equal to the maximum directional derivative of at . The gradient is an extension of the idea of derivative to functions with more than one variable.

Stated another way, a gradient is a vector that has orthogonal (coordinate) components that consist of the partial derivatives of a function with respect to each of its variables.

More precisely, we define the gradient, of to be the vector field: consisting of the various partial derivatives of . If is a unit vector in , then, by the chain rule, the directional derivative of in the direction of is simply the dot product: Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction is maximal in the direction of the gradient, and equal to for a unit vector in the direction of the gradient.

If is a differentiable function with smooth level sets , then the gradient vector field is perpendicular to the level sets of . For fix a level set , and let be a vector tangent to at . Then we can find a curve on with . Now since is a level set. Taking derivatives of both sides and applying the chain rule, we get that Thus, is perpendicular to at , i.e., the gradient of is perpendicular to the level sets of .