Gradient - Definition

Definition

The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, ..., xn) is denoted ∇f or where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,

In a rectangular coordinate system, the gradient is the vector field whose components are the partial derivatives of f:

where the ei are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

In the three-dimensional Cartesian coordinate system, this is given by

\frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}

where i, j, k are the standard unit vectors. For example, the gradient of the function

is:

\nabla f= \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} = 2\mathbf{i}+ 6y\mathbf{j} -\cos(z)\mathbf{k}.

In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system.

Read more about this topic:  Gradient

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