List of formulas related to the divergence of a vector field:
- Definition of divergence:
For a vector field (\mathbf{F} = (F_1, F_2, F_3)) in three-dimensional space, the divergence is defined as:
[
\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
] - Vector field in two-dimensional space:
For a vector field (\mathbf{F} = (F_1, F_2)) in two-dimensional space:
[
\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}
] - Physical meaning of divergence:
The divergence of a vector field at a point measures how much of the field “originates” from that point. A positive value indicates a source, and a negative value indicates a sink. - The formula for divergence in cylindrical coordinates is:
If (\mathbf{F} = (F_r, F_\theta, F_z)), then:
[
\nabla \cdot \mathbf{F} = \frac{1}{r} \frac{ \partial}{\partial r}(r F_r) + \frac{1}{r} \frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}
] - The formula for divergence in spherical coordinates is:
If (\mathbf{F} = (F_r, F_\theta, F_\phi)), then:
[
\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_\phi}{\partial \phi}
]