Fundamental Theorems
Gradient Theorem
Green's Theorem
Stoke's Theorem
Divergence Theorem or Gauss Theorem
In order to transform an equation from the differential to the integral form (vice versa), the Gauss Theorem should be applied.It's important for fluid dynamics.
\(\boldsymbol{V}\) represents a volume in three-dimensional space of boundary \(\boldsymbol{S}\), \(\mathbf{n}\) is the outward pointing unit vector normal to \(\boldsymbol{S}\). If \(\mathbf{v}\) is a vector field defined on \(\boldsymbol{V}\), then the divergence theorem states that
\[ \oint_{\boldsymbol{S}} \mathbf{v} \bullet \mathbf{n} \mathrm{d} S=\int_{\boldsymbol{V}}(\nabla \bullet \mathbf{v}) \mathrm{d} V \]
- Implying that the net flux of a vector field through a closed surface is equal to the total volume of all sources and sinks (i.e., the volume integral of its divergence) over the region inside the surface.