Gradient Vector

The gradient is a vector operation in multivariable calculus that represents the direction and rate of maximum change of a scalar function at a given point in its domain. It is denoted by the symbol ∇ (called “nabla”) and acts on a scalar function f(x, y, z) to produce a vector-valued function.

The gradient of a scalar function f(x, y, z) is given by:

∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

where i, j, and k are the standard unit vectors in the x, y, and z directions, respectively, and ∂f/∂x, ∂f/∂y, and ∂f/∂z are the partial derivatives of f with respect to x, y, and z.

The gradient points in the direction of the steepest increase of the scalar function, and its magnitude represents the rate of change of the function in that direction. The gradient is orthogonal to the level surfaces of the function, which are the surfaces where the function takes a constant value.

Gradients play an essential role in various fields, such as optimization, machine learning, and physics. In particular, the gradient is central to the study of vector fields and their associated concepts, such as divergence, curl, and Stokes’ theorem.