1 Divergence
1.1 Intuition
In vector calculus divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector fields source at each point. Specifically, the divergence represents the volume density of the outward flux of the vector field from an infinitesimal volume around a given point.
Intuitively, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point.
1.2 General Definition
The divergence of a vector field \(\boldsymbol{F}(\boldsymbol{x})\) at a point \(\boldsymbol{x}_{0}\) is defined as the limit of the ratio of the surface integral of \(\boldsymbol{F}\) out of the closed surface of a volume \(V\) enclosing \(\boldsymbol{x}_{0}\) to the volume of \(V\) as \(V\) shrinks to zero
\[ \text{div}\boldsymbol{F}|_{\boldsymbol{x_{0}}}=\lim_{ V \to 0 } \frac{1}{\lvert V \rvert }\iint_{S(V)}\boldsymbol{F}\cdot\hat{\boldsymbol{n}}~dS, \]
where \(\lvert V \rvert\) is the volume of \(V\), \(S(V)\) is boundary of \(V\), and \(\hat{ \boldsymbol{n}}\) is the outward unit normal to that surface.
It can be shown that the above limit always converges to the same value for any sequence of volumes that contain \(\boldsymbol{x}_{0}\) and approach zero volume. The result, \(\text{div}(\boldsymbol{F})\) is a scalar function of \(\boldsymbol{x}\).
A vector field with zero divergence everywhere is called [[Solenoid|solenoidal]], in which case any closed surface has no net flux across it.
1.3 Divergence in Different Coordinate Systems
The following are a collection of definitions of divergence under difference coordinate systems including cartesian, cylindrical, spherical and tensor field systems.
In \(n\)-dimensional Cartesian coordinates the divergence of a continuously differentiable vector field \(\boldsymbol{F}=F_{x_{1}}\boldsymbol{i}_{1}+\dots +F_{x_{n}}\boldsymbol{i}_{n}\) is defined as the scalar-value function
\[ \text{div}\boldsymbol{F}=\Delta \cdot \boldsymbol{F}=\left( \frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}} \right)\cdot (F_{x_{1}}, \dots, F_{x_{n}})= \frac{{\partial F_{x_{1}}}}{\partial x_{1}}+\dots+\frac{{\partial F_{x_{n}}}}{\partial x_{n}}. \]