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\text{[math]}

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 $\text{[math]}$ of a differentiable function $\text{[math]}$ which takes real values. Specifically, the gradient at $\text{[math]}$ is a vector in $\text{[math]}$ which points in the direction in which $\text{[math]}$ increases most rapidly at $\text{[math]}$ . The magnitude of the gradient at $\text{[math]}$ is equal to the maximum directional derivative of $\text{[math]}$ at $\text{[math]}$ . 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 $\text{[math]}$ , then

\text{[math]}

Observe that in this case, the gradient vector $\text{[math]}$ is orthogonal to the "level curve" defined by $\text{[math]}$ , which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of $\text{[math]}$ , 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, $\text{[math]}$ of $\text{[math]}$ to be the vector field:

$\text{[math]}$

consisting of the various partial derivatives of $\text{[math]}$ . If $\text{[math]}$ is a unit vector in $\text{[math]}$ , then, by the chain rule, the directional derivative of $\text{[math]}$ in the direction of $\text{[math]}$ is simply the dot product:

$\text{[math]}$

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction $\text{[math]}$ is maximal in the direction of the gradient, and equal to $\text{[math]}$ for $\text{[math]}$ a unit vector in the direction of the gradient.

Properties of the Gradient[edit]

If $\text{[math]}$ is a differentiable function with smooth level sets $\text{[math]}$ , then the gradient vector field $\text{[math]}$ is perpendicular to the level sets of $\text{[math]}$ . For fix a level set $\text{[math]}$ , and let $\text{[math]}$ be a vector tangent to $\text{[math]}$ at $\text{[math]}$ . Then we can find a curve $\text{[math]}$ on $\text{[math]}$ with $\text{[math]}$ . Now

$\text{[math]}$

since $\text{[math]}$ is a level set. Taking derivatives of both sides and applying the chain rule, we get that

$\text{[math]}$

Thus, $\text{[math]}$ is perpendicular to $\text{[math]}$ at $\text{[math]}$ , i.e., the gradient of $\text{[math]}$ is perpendicular to the level sets of $\text{[math]}$ .

Gradient

Properties of the Gradient[edit]

See also[edit]