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:



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 .