gradient - Interpretations of the gradient, Formal definition, Linear approximation to a function, The gradient on manifolds

A measure of the inclination of a straight line to a fixed straight line. In mathematical terms, the gradient of a straight line, in a rectangular co-ordinate system, is the tangent of the angle made by the straight line and the positive x-axis. The gradient of a curve at a point P is the gradient of the tangent to the curve at the point P.

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

A generalization of the gradient, for functions which have vectorial values, is the Jacobian.

Interpretations of the gradient

Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). The gradient of H at a point is in the direction of the steepest slope or grade at that point.

The gradient can also be used to tell how things change in other directions rather than the direction of largest change. The gradient of the hill height function H dotted with a unit vector gives the slope of the surface in the direction of the vector.

The gradient is irrotational (and vice versa) and thus line integrals through a gradient field are path independent and can be evaluated with the gradient theorem.

Formal definition

The gradient of a scalar function f(x) with respect to a vector variable is denoted by where (nabla) denotes the vector differential operator del.

By definition, the gradient is a column vector whose components are the partial derivatives of f. For example, the gradient of the function

is:

Linear approximation to a function

The gradient of a function f from the Euclidean space R to R characterizes the best linear approximation to that function at any particular point x0 in R. The approximation is as follows:

for x close to x0, where is the gradient computed at x0.

The gradient on manifolds

For any differentiable function f on a Riemannian manifold M, the gradient of f is the vector field such that for any vector ξ,

where denotes the inner product on M (the metric) and ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart, ξf(p) will be:

The gradient of a function is related to the exterior derivative, since ξf(p) = df(ξ).