Volume dissipation

The volume of derivative is a concept of mathematical sub-region of vector analysis , in particular in the engineering is used. Volume derivation means the coordinate-free representation of the differential operators , which are important for vector analysis, gradient , divergence and rotation . The representation by means of the volume derivative is also used as a definition of these differential operators, depending on the subject area. Using the integral theorems of Gauss and Stokes it can be shown that this coordinate-free representation agrees with the other common definitions of these operators.

Operators of vector analysis as volume derivation

gradient

Let be a space area with volume and a scalar field. Then the gradient of the scalar field in the point can through ${\displaystyle {\mathcal {V}}}$ ${\displaystyle V}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle p\in {\mathcal {V}}}$

${\displaystyle \mathrm {grad} \,f(p)=\lim _{n\rightarrow \infty }{\frac {1}{V_{n}}}\oint _{\partial {\mathcal {V}}_{n}}f\mathrm {d} {\vec {A}}}$

be calculated. It is a surface integral formed with the vectorial outer surface element of addition referred to a sequence of space areas , with and where the corresponding volume designated. ${\displaystyle \oint _{\partial {\mathcal {V}}}}$ ${\displaystyle \mathrm {d} {\vec {A}}}$ ${\displaystyle \partial {\mathcal {V}}.}$ ${\displaystyle ({\mathcal {V}}_{n})}$ ${\displaystyle {\mathcal {V}}_{n}\subset {\mathcal {V}}}$ ${\displaystyle p\in {\mathcal {V}}_{n}}$ ${\displaystyle \lim _{n\to \infty }V_{n}=0}$ ${\displaystyle V_{n}}$

The facts are usually a little shorter through

${\displaystyle \mathrm {grad} \,f(p)=\lim _{V\rightarrow 0}{\frac {1}{V}}\oint _{\partial {\mathcal {V}}}f\mathrm {d} {\vec {A}}}$

written down.

divergence

Be a vector field . With the notation from the previous section, the divergence of the vector field in the point can be determined ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle p\in {\mathcal {V}}}$

${\displaystyle \mathrm {div} \,{\vec {F}}(p)=\lim _{V\rightarrow 0}{\frac {1}{V}}\oint _{\partial {\mathcal {V}}}{\vec {F}}\mathrm {d} {\vec {A}}}$

be calculated.

rotation

Be also a vector field. With the notation from the previous section, the rotation of the vector field in the point can be carried out ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle p\in {\mathcal {V}}}$

${\displaystyle \mathrm {rot} \,{\vec {F}}(p)=\lim _{V\rightarrow 0}{\frac {1}{V}}\oint _{\partial {\mathcal {V}}}\mathrm {d} {\vec {A}}\times {\vec {F}}}$

be calculated.

Volume derivation concept

A general definition of volume derivative is seldom given in the literature. Rather, as here in the article, it is also introduced as the coordinate-free representation of the three differential operators of vector analysis. When calculating a volume derivative of a function in spatial space in the point , a spatial area is selected with the content that contains the point . An approximation for the value of the volume derivative then results from the surface integral of over the edge of divided by By shrinking from to , the volume derivative then results as a limit value. ${\displaystyle g}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle V}$ ${\displaystyle p}$ ${\displaystyle g}$ ${\displaystyle \partial {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle V.}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle p}$

Sometimes, however, the equation also becomes

${\displaystyle f(p)=\lim _{V\to 0}{\frac {1}{V}}\int _{\mathcal {V}}f(x)dx}$

for a constant function called volume derivative. Using this representation and certain special cases of Gauss’s integral theorem, the above volume derivatives can be proven. This volume integral does not deal with the change in the function , but delivers its value at the point ${\displaystyle p\in {\mathcal {V}}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle p.}$

Similarity to the common derivation

In order to emphasize the relationship between the volume derivative and the ordinary derivative, the (ordinary) derivative of a scalar-valued function can also be used at the point through the boundary integral ${\displaystyle f'(p)}$ ${\displaystyle f}$ ${\displaystyle p}$

${\displaystyle f'(p)=\lim _{L\rightarrow 0}{\frac {1}{L}}\oint _{\partial {\mathcal {L}}}\mathrm {d} f={\frac {\mathrm {d} f}{\mathrm {d} x}}(p)}$

be noted. The interval including the value denotes the content (= the length) of and the edge of , i.e. its lower and upper limit. Shrinking from to results in the limit value. ${\displaystyle {\mathcal {L}}}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle L}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle \partial {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle p}$ ${\displaystyle f'(p)}$

Generalization through the Cartan derivation

In the transition to the more modern Cartan calculus, scalar and vector fields are replaced by differential forms that represent them: A scalar field can be viewed directly as a differential 0-form, via (whereby the canonical scalar product denotes and is open) can also be used as a 3-form. A vector field can act as a 1-form via and as a 2-form via . The relationship will each have the Hodge operator made: . Which translation takes place depends largely on the intended use of the scalar or vector field. In the following denote a -dimensional submanifold or -chain. It then always applies ${\displaystyle C^{\infty }(X)\to \Omega ^{3}(X),\Phi \mapsto (\omega _{\Phi }^{3}:{\vec {x}}\mapsto (\omega _{\Phi ,{\vec {x}}}^{3}:({\vec {\xi }},{\vec {\eta }},{\vec {\zeta }})\mapsto \Phi ({\vec {x}})\cdot \langle {\vec {\xi }},{\vec {\eta }}\times {\vec {\zeta }}\rangle ))}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle X\subset \mathbb {R} ^{3}}$ ${\displaystyle V(X)\to \Omega ^{1}(X),{\vec {F}}\mapsto ({\vec {x}}\mapsto \omega _{{\vec {F}},{\vec {x}}}^{1}:{\vec {\xi }}\mapsto \langle {\vec {F}}({\vec {x}}),{\vec {\xi }}\rangle )}$ ${\displaystyle V(X)\to \Omega ^{2}(X),{\vec {F}}\mapsto (\omega _{{\vec {F}},{\vec {x}}}^{2}:({\vec {\xi }},{\vec {\eta }})\mapsto \langle {\vec {F}}({\vec {x}}),{\vec {\xi }}\times {\vec {\eta }}\rangle )}$ ${\displaystyle \Phi =\star \omega _{\Phi }^{3},\omega _{\vec {F}}^{1}=\star \omega _{\vec {F}}^{2}}$ ${\displaystyle G_{m}}$ ${\displaystyle m}$ ${\displaystyle m}$

${\displaystyle \int _{G_{1}}{\vec {F}}\cdot d{\vec {s}}=\int _{G_{1}}\omega _{\vec {F}}^{1},\;\int _{G_{2}}{\vec {F}}\cdot d{\vec {A}}=\int _{G_{2}}\omega _{\vec {F}}^{2},\;\int _{G_{3}}\Phi \,dV=\int _{G_{3}}\omega _{\Phi }^{3}}$

The Cartan derivation generalizes the concept of the volume derivation of vector fields for shapes on manifolds of any dimension. Let it denote the parallelepiped spanned by these vectors (considered as a chain, whereby in the case of a manifold the tangent vectors are the same tangent space) as well as the integral of a form over the edge of this parallelepiped. For every -form there is always a unique -form , which corresponds to the linear part of the integral over the edge of every parallelepiped if it becomes infinitesimal (i.e. ): ${\displaystyle \Pi ({\vec {\xi }}_{1},\dots ,{\vec {\xi }}_{k+1})}$ ${\displaystyle (k+1)}$ ${\displaystyle {\vec {\xi }}_{i}}$ ${\displaystyle F_{\omega }({\vec {\xi }}_{1},\dots ,{\vec {\xi }}_{k+1}):=\int _{\partial \Pi (\dots )}\omega }$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \omega }$ ${\displaystyle (k+1)}$ ${\displaystyle d\omega }$ ${\displaystyle \epsilon \to 0}$

${\displaystyle F_{\omega }(\epsilon {\vec {\xi }}_{1},\dots ,\epsilon {\vec {\xi }}_{k+1})=\epsilon ^{k+1}d\omega ({\vec {\xi }}_{1},\dots ,{\vec {\xi }}_{k+1})+o(\epsilon ^{k+1}).}$

If so , it is the same as the ordinary differential. The Cartan derivation generalizes the operations gradient, divergence and rotation in the following way: ${\displaystyle k=1}$

${\displaystyle \omega _{\operatorname {grad} \Phi }^{1}\equiv d\Phi ,\;\omega _{\operatorname {rot} {\vec {F}}}^{2}\equiv d\omega _{\vec {F}}^{1},\;\omega _{\operatorname {div} {\vec {F}}}^{3}\equiv d\omega _{\vec {F}}^{2}}$

literature

• IN Bronstein, KA Semendjajew: Pocket book of mathematics . 6th edition. Verlag Harri Deutsch, Frankfurt am Main 2006, ISBN 978-3-8171-2006-2 .

• K. Simonyi: Theoretical electrical engineering . 9th edition. VEB Deutscher Verlag der Wissenschaften, Berlin 1989.

• GE Joos, E. Richter: Higher Mathematics . 13th edition. Nikol-Verlag, Hamburg 2012.

Individual evidence

1. ↑ Horst Stöcker: Mathematics - The Basic Course: Analysis for Engineering Students, Volume 2 . Harri Deutsch, 1996, ISBN 3-8171-1340-4 , pp. 173 .
2. ↑ Horst Stöcker: Mathematics - The Basic Course: Analysis for Engineering Students, Volume 2 . Harri Deutsch, 1996, ISBN 3-8171-1340-4 , pp. 173-174 .
3. ↑ Horst Stöcker: Mathematics - The Basic Course: Analysis for Engineering Students, Volume 2 . Harri Deutsch, 1996, ISBN 3-8171-1340-4 , pp. 174 .
4. ^ E. Zeidler (ed.): Springer-Taschenbuch der Mathematik / greed. By IN Bronstein and KA Semendjaew. Continued by G. Grosche . Springer Spectrum, Wiesbaden 2013, ISBN 978-3-8351-0123-4 , p. 378 .
5. ↑ VI Arnol'd: Mathematical Methods of Classical Mechanics, Second Edition . Ed .: S. Axler, FW Gehring, KA Ribet. Springer Science + Business Media, New York 1989, ISBN 0-387-96890-3 , pp. 190 .