Gauss’s Theorem

In the derivation of the conservation equations, Gauss’s theorem is frequently used. This theorem relates integrals over a domain to an integral over the boundary of this domain. It can be used to relate a volume integral to a surface integral or an area integral to a contour integral.

Gauss’s theorem states that when \(f(\mathbf{x})\) is piecewise continuously differentiable, that is, a \(C^{0}\) function, then

\[\int\limits_{\Omega}\cfrac{\partial f(\mathbf{x},t)}{\partial x_{i}}\text{d}\Omega=\int\limits_{\Gamma} n_{i}f(\mathbf{x}) d \Gamma\]

The Divergence Theorem-vector

Consider an arbitrary differentiable vector field \(v(\mathbf{x},t)\) defined in some finite region of physical space. Let \(V\) be a volume in this space with a closed surface S bounding the volume, and let the outward normal to this bounding surface be \(\mathbf{n}\). The divergence theorem of Gauss states that (in symbolic and index notation)

\[\int\limits_{S}\mathbf{v}\cdot\mathbf{n}\text{d}S=\int\limits_{V}\text{div }\mathbf{v}\text{d}V\]

\[\begin{split}\int\limits_{S}{v}_{i}\cdot{n}_{i}\text{d}S=\int\limits_{V}\cfrac{\partial {v}_{i}}{\partial {x}_{i}}\text{d}V\\\end{split}\]

The divergence theorem is also called Gauss’s theorem, named after the German mathematician Johann Carl Friedrich Gauss (1777–1855). The divergence theorem allows us to transform a volume integral of the divergence of a vector into an area integral over the surface that defines the volume. For any vector \(\vec{G}\), the divergence of \(\vec{G}\) is defined as \(\nabla \cdot \vec{G}\), and the divergence theorm is written as

\[\int_{V}\nabla\cdot\vec{G}=\oint_{A}\vec{G}\cdot\vec{n}dA\]

\[\begin{split}\vec{G}=g_{1}(x,y,z)\vec{i}+g_{2}(x,y,z)\vec{j}+g_{3}(x,y,z)\vec{k} =\begin{bmatrix} g_{1}(x,y,z)\\g_{2}(x,y,z)\\g_{3}(x,y,z) \end{bmatrix}\\\end{split}\]

\[\nabla \cdot \vec{G}=\cfrac{\partial g_{1}}{\partial x}+\cfrac{\partial g_{2}}{\partial y}+\cfrac{\partial g_{3}}{\partial z}\]

\[\vec{G}\cdot\vec{n}=g_{1}n_{1}+g_{2}n_{2}+g_{3}n_{3}=g_{1}n_{x}+g_{2}n_{y}+g_{3}n_{z}\]

\[\begin{split}\vec{n}=n_{1}\vec{i}+n_{2}\vec{j}+n_{3}\vec{k}=n_{x}\vec{i}+n_{y}\vec{j}+n_{z}\vec{k} =\begin{bmatrix} n_{1}\\n_{2}\\n_{3} \end{bmatrix} =\begin{bmatrix} n_{x}\\n_{y}\\n_{z} \end{bmatrix}\\\end{split}\]

The Divergence Theorem-tensors

Consider an arbitrary differentiable tensor field \(T_{ij\cdots k}(\mathbf{x},t)\) defined in some finite region of physical space. Let \(S\) be a closed surface bounding a volume \(V\) in this space, and let the outward normal to \(S\) be \(\mathbf{n}\). The divergence theorem of Gauss then states that

\[\begin{split}\int\limits_{S}T_{ij\cdots k}\cdot{n}_{k}\text{d}S=\int\limits_{V}\cfrac{\partial T_{ij\cdots k}}{\partial {x}_{k}}\text{d}V\\\end{split}\]

For a second order tensor,

\[\begin{split}\int\limits_{S}\mathbf{T}\cdot\mathbf{n}\text{d}S=\int\limits_{V}\text{div }\mathbf{T}\text{d}V\\\end{split}\]

\[\begin{split}\int\limits_{S}{T}_{ij}\cdot{n}_{j}\text{d}S=\int\limits_{V}\cfrac{\partial {T}_{ij}}{\partial {x}_{j}}\text{d}V\\\end{split}\]