|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 coordinate components that consist of the partial derivatives of a function with respect to each of its variables. For example, if , then
Observe that in this case, the gradient vector is orthogonal to the "level curve" defined by , which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of , and vectors outward from the origin are perpendicular to circles centered at the origin. We'll see later that this is a case of a more general property of the gradient.
More precisely, we define the gradient, of to be the vector field:
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.
Properties 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 .