Overview
Vector operators are functions that take a vector as an input, and either return another vector, or a scalar. The most common vector space over which the vector operators are defined is {% \mathbb{R}^3 %}, mainly because of its prominence in classical physics.
Some operators are only defined for {% \mathbb{R}^3 %}.
Notation
A point in {% \mathbb{R}^3 %} is often written as
{% \vec{r} = x \hat{x} + y \hat{y} + z\hat{z} %}
The nabla symbol {% \nabla %} is interpreted to mean
{% \nabla = \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z} %}
which can be used to defined several operators.
Operators
Given a vector field {% \vec{V} (\vec{r}) = \vec{V}(x,y,z) %}
- Gradient
{% \nabla \vec{V} = ( \frac{\partial V_x}{\partial x} , \frac{\partial V_y}{\partial y} , \frac{\partial V_z}{\partial z} ) %}
- Divergence
{% \nabla \cdot \vec{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} %}
- Curl
{% \nabla \times \vec{V} = (\frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z}) \hat{x} + (\frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x}) \hat{y} + (\frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}) \hat{z} %}
- Laplacian (Laplace Operator)
{% \nabla ^2 A = \frac{\partial^2 A_x}{\partial x^2} + \frac{\partial^2 A_y}{\partial y^2} + \frac{\partial^2 A_z}{\partial z^2} %}
Coordinate Representations
The definitions of the vector operators given above are given in Cartesian space. The following give the definitions of the operators in different coordinate systems.