# Gradient

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In mathematics, the **gradient** is a vector associated to a point *p* of a differentiable function *f*(*x*_{1},...,*x*_{n}) which takes real values. Specifically, the gradient at *p* is a vector in *R*^{n} which points in the direction in which *f* increases most rapidly at *p*. The magnitude of the gradient at *p* is equal to the maximum directional derivative of *f* at *p*. 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 *f*(*x*,*y*) = *x*^{2} + *y*^{2}, then

- . Observe that in this case, the gradient vector (2
*x*,2*y*) is orthogonal to the "level curve" defined by*x*^{2}+*y*^{2}=*r*^{2}, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of*f*, 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.

- . Observe that in this case, the gradient vector (2

More precisely, we define the gradient, of *f* to be the vector field:

consisting of the various partial derivatives of *f*. If *u* is a unit vector in *R*^{n}, then, by the chain rule, the directional derivative of *f* in the direction of *u* is simply the dot product:

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction *u* is maximal in the direction of the gradient, and equal to for *u* a unit vector in the direction of the gradient.

## Properties of the Gradient

If *f* is a differentiable function with smooth level sets *f*^{ − 1}(*c*), then the gradient vector field is perpendicular to the level sets of *f*. For fix a level set *S* = *f*^{ − 1}(*c*), and let *v* be a vector tangent to *S* at *p*. Then we can find a curve γ(*t*) on *S* with γ'(0) = *v*. Now

since *S* is a level set. Taking derivatives of both sides and applying the chain rule, we get that

Thus, is perpendicular to *v* at *p*, i.e., the gradient of *f* is perpendicular to the level sets of *f*.