gradientfield

A gradient field is a vector field that arises as the gradient of a scalar function. If f is a differentiable function defined on an open set U in Euclidean space R^n, then the gradient of f, denoted ∇f, is the vector field whose components are the partial derivatives ∂f/∂x1, ∂f/2, ..., ∂f/∂xn. The gradient at a point gives the direction of steepest ascent of f, and its magnitude equals the maximal rate of increase of f at that point.

Properties and interpretation: A gradient field F = ∇f is a conservative field, meaning line integrals along a curve depend only on the endpoints. Consequently, the work done by F along any path between two points is path-independent, and F can be written as the gradient of its potential function f. In simply connected domains, a vector field with zero curl is a gradient field; conversely, a gradient field automatically has zero curl where the derivatives are smooth.

Examples: In the plane, the gradient of f(x,y) = x^2 + y^2 is ∇f = (2x, 2y). For f(x,y) = xy, ∇f = (y, x). In three dimensions, ∇(x^2 + y^2 + z^2) = (2x, 2y, 2z).

Applications: Gradient fields are central in physics and engineering as potential or force fields, in optimization through gradient descent, and in differential geometry with respect to a given metric. They also provide the fundamental link between a scalar potential and the vector field describing its spatial variation. A given vector field is a gradient field only under suitable smoothness and topological conditions.