Divergence
Table of Contents
The divergence measures the rate at which the vector field \(\small \mathbf{F}\) spreads out or converges at a given point in space.
Cartesian Coordinates #
The divergence of a vector field \(\small \mathbf{F}\) in cartesian coordinates is given by:
$$ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$
Where:
- \(\small x, y, z\) are the usual cartesian coordinates,
- \(\small \nabla\) is the del operator (also known as the gradient operator),
- \(\cdot\) denotes the dot product, and
- \(\partial\) denotes the partial derivative
Cylindrical Coordinates #
The divergence of a vector field \(\small \mathbf{F}\) in cylindrical coordinates is given by:
$$ \nabla \cdot \mathbf{F} = \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho F_\rho) + \frac{1}{\rho} \frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} $$
Where:
- \(\small \rho\) is the radial coordinate, \(\small \phi\) is the azimuthal coordinate, and \(\small z\) is the vertical coordinate.
Spherical Coordinates #
The divergence of a vector field \(\small \mathbf{F}\) in spherical coordinates is given by:
$$ \nabla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 F_r) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta F_\theta) + \frac{1}{r \sin \theta} \frac{\partial F_\varphi}{\partial \varphi} $$
Where:
- \(\small r\) is the radial coordinate, \(\small \theta\) is the polar angle, and \(\small \varphi\) is the azimuthal angle.