Fundamental Theorems
An overview of the integral theorems that connect local differential quantities with global integral quantities in fluid dynamics.
Gradient Theorem
Green’s Theorem
Stokes’ Theorem
Divergence Theorem or Gauss Theorem
To transform an equation between differential and integral forms, the Gauss theorem can be applied. It is fundamental to 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.