Definition:Gradient Operator

Definition
Let $\mathbf V$ be a vector space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.

Let $\map U {x_1, x_2, \ldots, x_n}: \mathbf V \to \R$ be a real-valued function on $\mathbf V$.

The gradient of $U$ is defined as:

Thus the gradient is a vector in $\mathbf V$.

Real Cartesian Space
The usual context in which the gradient operator is encountered is real Cartesian space:

Also known as
The gradient of $U$ is usually vocalised grad $U$.