---
tag: 0H6DP8CHND
kind: concept
name: Del operator
aliases: [nabla, del]
msc: [26B12]
uses: [0ABK9V9Y64, 0MWV6ZTK6S, 0NF6GQNXMY, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Del
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Del
    locator: Notational uses
same_as:
  wikidata: Q334508
  wikipedia: Del
---

## 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|0ABK9V9Y64]] $\nabla f$, the [[divergence|0MWV6ZTK6S]] $\nabla \cdot F$, the [[curl|0NF6GQNXMY]] $\nabla \times F$ in $\mathbb{R}^3$, and the [[Laplacian|0E0B6RA8K0]] $\nabla^2 f = \nabla \cdot \nabla f$.

## 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.