Gradient

<math>\frac{d}{dx} \sin x=?\,</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 <math>p</math> of a differentiable function <math>f(x_1,...,x_n)</math> which takes real values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</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 <math>f(x,y) = x^2+y^2</math>, then

<math>\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x,2y)</math>.

Observe that in this case, the gradient vector <math>(2x,2y)</math> is orthogonal to the "level curve" defined by <math>x^2+y^2=r^2</math>, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of <math>f</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, <math>\nabla f</math> of <math>f</math> to be the vector field:

<math> \nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n}) </math>

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

<math> \nabla f \cdot u </math>

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.

Properties of the Gradient

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

<math> f\circ\gamma(t) = c </math>

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

<math> \nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0 </math>

Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.

See also

• Curl
• Divergence
• Laplacian