Gaussian integral theorem

The Gaussian integral theorem , also Gauss-Ostrogradski's theorem or divergence theorem , is a result of vector analysis . It establishes a connection between the divergence of a vector field and the flow through a closed surface given by the field .

The integral theorem named after Gauss follows as a special case from the Stokes theorem , which also generalizes the main theorem of differential and integral calculus .

Formulation of the sentence

In three dimensions, a region V is shown, which is bounded by the closed surface S = ∂ V , oriented by the outer surface normal vector n .

Let it be a compact set with a smooth edge in sections , let the edge be oriented by an outer normal unit vector field . Furthermore, let the vector field be continuously differentiable on an open set with . Then applies ${\displaystyle V\subset \mathbb {R} ^{n}}$ ${\displaystyle S=\partial V}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle U}$ ${\displaystyle V\subseteq U}$

${\displaystyle \int _{V}\operatorname {div} {\vec {F}}\;\mathrm {d} ^{(n)}V=\oint _{S}{\vec {F}}\cdot {\vec {n}}\;\mathrm {d} ^{(n-1)}S}$

where denotes the standard scalar product of the two vectors. ${\displaystyle {\vec {F}}\cdot {\vec {n}}}$

example

Is the closed unit ball in , then applies as well . ${\displaystyle V:=\{{\vec {x}}\in \mathbb {R} ^{3}\colon \|{\vec {x}}\|_{2}\leq 1\}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S=\partial V=\{{\vec {x}}\in \mathbb {R} ^{3}\colon \|{\vec {x}}\|_{2}=1\}}$ ${\displaystyle {\vec {n}}({\vec {x}})={\vec {x}}}$

The following applies to the vector field with . ${\displaystyle {\vec {F}}\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}$ ${\displaystyle {\vec {F}}({\vec {x}})={\vec {x}}}$ ${\displaystyle \operatorname {div} {\vec {F}}({\vec {x}})=3}$

It follows

${\displaystyle \int _{V}\operatorname {div} {\vec {F}}\;\mathrm {d} ^{3}V=\int _{V}3\;\mathrm {d} ^{3}V=3\cdot {\frac {4}{3}}\pi =4\pi }$

such as

${\displaystyle \oint _{S}{\vec {F}}\cdot {\vec {n}}\;\mathrm {d} ^{2}S=\oint _{S}{\vec {x}}\cdot {\vec {x}}\;\mathrm {d} ^{2}S=\oint _{S}1\;\mathrm {d} ^{2}S=4\pi \,.}$

In the calculation it was used that applies to all and that the three-dimensional unit sphere has the volume and the surface . ${\displaystyle {\vec {x}}\cdot {\vec {x}}=\|{\vec {x}}\|_{2}^{2}=1}$ ${\displaystyle {\vec {x}}\in S}$ ${\displaystyle {\tfrac {4}{3}}\pi }$ ${\displaystyle 4\pi }$

Inferences

Further identities can be derived from the Gaussian integral theorem. To simplify matters, the notation and as well as the Nabla notation are used below . ${\displaystyle \mathrm {d} {\vec {S}}:={\vec {n}}\;\mathrm {d} ^{(n-1)}S}$ ${\displaystyle \mathrm {d} V:=\mathrm {d} ^{(n)}V}$

• If one applies the Gaussian integral theorem to the product of a scalar field with a vector field , one obtains ${\displaystyle f}$ ${\displaystyle {\vec {G}}}$

${\displaystyle \int _{V}\left(\left(\nabla f\right)\cdot {\vec {G}}+f\left(\nabla \cdot {\vec {G}}\right)\right)\mathrm {d} V=\int _{V}\nabla \cdot \left(f{\vec {G}}\right)\mathrm {d} V=\oint _{S}f{\vec {G}}\cdot \mathrm {d} {\vec {S}}\,.}$

If you look at the special case , you get the first Green identity . ${\displaystyle {\vec {G}}=\nabla g}$

If, on the other hand , one considers the special case , one obtains ${\displaystyle {\vec {G}}=\mathrm {const.} }$

${\displaystyle \int _{V}\left(\nabla f\right)\mathrm {d} V=\oint _{S}f\,\mathrm {d} {\vec {S}}}$

or, broken down by components,

${\displaystyle \int _{V}{\frac {\partial f}{\partial x_{i}}}\,\mathrm {d} V=\oint _{S}fn_{i}\,\mathrm {d} ^{\left(n-1\right)}S\,.}$

• If one applies the Gaussian integral theorem for to the cross product of two vector fields and , one obtains ${\displaystyle n=3}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {G}}}$

${\displaystyle \int _{V}\left({\vec {G}}\cdot \left(\nabla \times {\vec {F}}\right)-{\vec {F}}\cdot \left(\nabla \times {\vec {G}}\right)\right)\,\mathrm {d} V=\int _{V}\left(\nabla \cdot \left({\vec {F}}\times {\vec {G}}\right)\right)\,\mathrm {d} V=\oint _{S}\left({\vec {F}}\times {\vec {G}}\right)\cdot \mathrm {d} {\vec {S}}\,.}$

If you consider the special case , you get ${\displaystyle {\vec {G}}=\mathrm {const.} }$

${\displaystyle \int _{V}\left(\nabla \times {\vec {F}}\right)\,\mathrm {d} V=\oint _{S}\mathrm {d} {\vec {S}}\times {\vec {F}}\,.}$

• If one applies the Gaussian integral theorem to vector fields im , multiplies the integrals with basis vectors of the standard basis , uses the properties of the dyadic product " " and adds the results, one obtains the generalization to tensors : ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\hat {e}}_{1,2,...,n}}$ ${\displaystyle \otimes }$

${\displaystyle {\begin{array}{rcl}\displaystyle \oint _{S}{\vec {F}}_{i}\cdot \mathrm {d} {\vec {S}}&=&\displaystyle \int _{V}\nabla \cdot {\vec {F}}_{i}\,\mathrm {d} V=\int _{V}\operatorname {div} {\vec {F}}_{i}\,\mathrm {d} V\,,\quad i=1,2,...,n\\\displaystyle \rightarrow \sum _{i=1}^{n}\oint _{S}{\vec {F}}_{i}\cdot \mathrm {d} {\vec {S}}\;{\hat {e}}_{i}&=&\displaystyle \oint _{S}\sum _{i=1}^{n}({\hat {e}}_{i}\otimes {\vec {F}}_{i})\cdot \mathrm {d} {\vec {S}}\\&=&\displaystyle \sum _{i=1}^{n}\int _{V}\nabla \cdot {\vec {F}}_{i}\,\mathrm {d} V\;{\hat {e}}_{i}=\int _{V}\sum _{i=1}^{n}\nabla \cdot ({\vec {F}}_{i}\otimes {\hat {e}}_{i})\,\mathrm {d} V=\int _{V}\sum _{i=1}^{n}\operatorname {div} ({\vec {F}}_{i}\otimes {\hat {e}}_{i})\,\mathrm {d} V\\\rightarrow \displaystyle \oint _{S}\mathbf {T} ^{\top }\cdot \mathrm {d} {\vec {S}}&=&\displaystyle \int _{V}\operatorname {div} (\mathbf {T} )\,\mathrm {d} V\,.\end{array}}}$

The superscript stands for the transposition . However, another definition of the divergence of tensors occurs in the literature ${\displaystyle \top }$

${\displaystyle {\widetilde {\operatorname {div} }}(\mathbf {T} ):=\mathbf {T} \cdot \nabla =\nabla \cdot \mathbf {T} ^{\top }=\operatorname {div} (\mathbf {T} ^{\top })\,,}$

which differs from the present one by the transposition of the argument. With this divergence operator:

${\displaystyle \oint _{S}\mathbf {T} \cdot \mathrm {d} {\vec {S}}=\int _{V}{\widetilde {\operatorname {div} }}(\mathbf {T} )\,\mathrm {d} V\,.}$

• If one applies the Gaussian integral theorem to the derivative of a real function on the interval , then one obtains the main theorem of differential and integral calculus . The evaluation of the integral at the interval ends in the main clause corresponds to the evaluation of the edge integral in the divergence clause. ${\displaystyle n=1}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$

${\displaystyle \int _{[a,b]}{\frac {\mathrm {d} f}{\mathrm {d} x}}\,\mathrm {d} x=\oint _{\partial [a,b]}f\cdot \,\mathrm {d} {\vec {S}}=f(b)-f(a)\,.}$

Applications

Liquids, gases, electrodynamics

The theorem is used to describe the conservation of mass , momentum and energy in any volume: The integral of the source distribution (sum of the divergence of a vector field) over the volume inside an envelope multiplied by a constant gives the total flow (the envelope integral) of the total flow through the envelope of this volume.

Gravity

See also : Newton's shell theorem

In the gravitational field one obtains: The surface integral is -4π G times the mass inside, as long as the mass is distributed radially symmetrically in it (constant density at a given distance from the center) and independent of any masses outside (also radially symmetrically distributed). In particular, the following applies: The whole sphere outside a sphere has no (additional) influence as long as its mass is distributed radially symmetrically. Only the sum of the sources and sinks in the interior have an effect.

Partial integration in the multi-dimensional

The Gaussian integral theorem leads to a formula for partial integration in the multi-dimensional

${\displaystyle \int _{\Omega }\varphi \,\operatorname {div} \,{\vec {v}}\;\mathrm {d} V=\int _{\partial \Omega }\varphi \,{\vec {v}}\cdot \mathrm {d} {\vec {S}}-\int _{\Omega }{\vec {v}}\cdot \operatorname {grad} \,\varphi \;\mathrm {d} V}$.

meaning

The Gaussian integral theorem is used in many areas of physics, especially in electrodynamics and fluid dynamics .

In the latter case, the meaning of the sentence is particularly clear. Let us assume that the vector field describes flowing water in a certain area of ​​space. Then the divergence of just describes the strength of all sources and sinks in individual points. If you want to know how much water flows out of a certain area , it is intuitively clear that you have the following two options: ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle V}$

• One examines or measures how much water exits and enters through the surface of . This corresponds to the flow of vertical components on the surface as a surface integral. ${\displaystyle V}$
• One balances (measures) inside the limited volume, how much water disappears in total in sinks (holes) and how much comes from sources (water inflows). So you add the effects of sources and sinks. Alternatively and equally, this is then realized by the volume integral over the divergence. ${\displaystyle V}$

The Gaussian integral theorem states that in fact both possibilities always lead to the goal in an absolutely equivalent manner. It thus also has the character of a law of conservation of energy.

history

The sentence was probably formulated for the first time by Joseph Louis Lagrange in 1762 and was later rediscovered independently by Carl Friedrich Gauß (1813), George Green (1825) and Michail Ostrogradski (1831). Ostrogradski also provided the first formal proof.

literature

• Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in ⁿ and applications ${\displaystyle \mathbb {R} }$ , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 .
• Konrad Königsberger : Analysis 2 , Springer, Berlin 2004.
• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .