Gradient
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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.
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 .
