Del operator

Summary

The del operator $\nabla = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right)$ is a vector of partial derivative operators on $\mathbb{R}^n$. It gives a short notation for the gradient $\nabla f$, the divergence $\nabla \cdot F$, the curl $\nabla \times F$ in $\mathbb{R}^3$, and the Laplacian $\nabla^2 f = \nabla \cdot \nabla f$.

Remarks

The notation treats $\nabla$ like a vector, but it is an operator. For example, $\nabla \cdot F$ and $F \cdot \nabla$ are different: the first is a function, and the second is the operator $\sum_i F_i \, \partial / \partial x_i$. The directional derivative of $f$ along $v$ is $(v \cdot \nabla) f = v \cdot \nabla f$. The symbol $\nabla$ is called nabla.

Explanations

L1 human

Del is a shorthand that makes vector calculus formulas look like vector algebra. Multiply it by a function to get the gradient. Dot it with a field to get the divergence, and cross it with a field to get the curl.