Divergence

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The divergence of a vector field is a scalar field, with each point corresponding to the magnitude of the change in density of the vector field at said point. Divergence can be calculated by taking the dot product of the vector field and the del operator, although it is formally defined in $\mathbb R^{3}$ as

$\text{div}\ \vec{\mathbf F}=\lim_{V_{x,y,z}\to0}\frac{1}{V_{x,y,z}}\iint\vec{\mathbf F}\cdot d\vec s$

Divergence can be thought of as flux density. A vector field which has a divergence of zero is called an incompressible vector field.

Given the function

$\vec{\mathbf v}=f_x\mathbf{\hat i}+f_y\mathbf{\hat j}+f_z\mathbf{\hat k}$

divergence is equal to

$\text{div}(\vec{\mathbf v})=\nabla\cdot\vec{\mathbf v}=\frac{\part}{\part x}f_x+\frac{\part}{\part y}f_y+\frac{\part}{\part z}f_z$

In $n$ dimensions, divergence of $\vec{ \mathbf{v} }$ is equal to

$\text{div}(\vec{\mathbf v})=\frac{\part}{\part x_0}f_0+\cdots+\frac{\part}{\part x_n}f_n$

See also

• Gradient
• Curl
• Divergence theorem