Volume integral

A volume integral or triple integral is a special case of multi-dimensional integral calculus in mathematics , which is mainly used in physics . It extends the surface integral to the integration over any three-dimensional integration area, whereby a function is integrated three times in a row, each over one direction of a three-dimensional space . However, it does not necessarily have to be a volume of a geometric body . To simplify the representation, often only a single integral symbol is written and the volume integration is only indicated by the volume element : ${\displaystyle \mathrm {d} V}$

${\displaystyle \iiint _{V}f(r)\mathrm {\,} d^{3}r=\displaystyle \int _{V}f({\vec {x}})\,\mathrm {d} V}$,

The function to be integrated depends on at least three variables for a (Cartesian) description in three-dimensional space , but higher-dimensional spaces are also possible. Note that the integration area appears here in two meanings, once in the volume element and once as an identifier for the volume over which the integration takes place. ${\displaystyle {\vec {x}}=(x,y,z)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle V}$ ${\displaystyle \mathrm {d} V}$

Terms

It is a scalar volume integral if the integrand and the volume element are scalar. For a vector integrand, e.g. B. a vector field , the volume element is also a vector , so that a vector volume integral results. ${\displaystyle f}$ ${\displaystyle \mathrm {d} V}$ ${\displaystyle {\vec {f}}}$ ${\displaystyle \mathrm {d} {\vec {V}}}$

The integration area is three-dimensional, e.g. B. an integration volume . The differential of the volume integral, e.g. B. , is also three-dimensional and can clearly be understood as an infinitesimal , infinitely small volume. In clear terms, the volume integral sums up all function values ​​of each volume element. This technique is used in physics, for example, to calculate the mass of a body with an unevenly distributed density . If one sets , the volume of the integration area results itself. ${\displaystyle V}$ ${\displaystyle \mathrm {d} V}$ ${\displaystyle f}$ ${\displaystyle f=1}$

Parameterization

In order to calculate a volume integral, a parameterization of the integration area is usually necessary. Finding a suitable substitution function is not trivial. It often transforms the volume integral from one coordinate system to another in order to simplify the calculation or to enable it at all. A parameterization describes how the integration area, which is limited by the edge , is traversed during integration. For a three-dimensional integration area, three parameters are necessary to describe the parameterized volume. Since the integrand also depends on at least three variables, the parameterization must be a vector-valued function . If such a parameterization is given, then the (vectorial) volume integral can be transformed according to the transformation theorem as follows: ${\displaystyle \partial V}$ ${\displaystyle (u,v,w)}$ ${\displaystyle {\vec {\xi }}(u,v,w)}$

${\displaystyle \iiint _{V}{\vec {f}}({\vec {x}})\mathrm {d} {\vec {V}}=\iiint _{\partial V}{\vec {f}}\left({\vec {\xi }}(u,v,w)\right)\cdot {\vec {N}}\mathrm {d} (u,v,w)}$

On the right-hand side of the equation, the parameterization function takes the place of the arguments of the integrand. The point indicates a scalar multiplication of the function to be integrated with the (vectorial) volume element . The volume element consists of a new (parameterized) volume differential and a vector factor. Based on the fact that three lengths multiplied together result in a volume, it is usually simple ${\displaystyle {\vec {\xi }}(u,v,w)}$ ${\displaystyle \mathrm {d} {\vec {V}}={\vec {N}}\,\mathrm {d} (u,v,w)}$

${\displaystyle \mathrm {d} (u,v,w)=\mathrm {d} u\mathrm {d} v\mathrm {d} w}$

set (according to the volume formula of a cuboid ). Other complex volume formulas could also be of interest for a numerical calculation.

The vector factor is the late product of all partial derivatives of : ${\displaystyle {\vec {\xi }}(u,v,w)}$

${\displaystyle {\vec {N}}=\left({\dfrac {\partial {\vec {\xi }}}{\partial u}}\times {\dfrac {\partial {\vec {\xi }}}{\partial v}}\right)\cdot {\dfrac {\partial {\vec {\xi }}}{\partial w}}}$

In general, late products can also be written as determinants , so the following applies here:

${\displaystyle {\vec {N}}=\left({\dfrac {\partial {\vec {\xi }}}{\partial u}}\times {\dfrac {\partial {\vec {\xi }}}{\partial v}}\right)\cdot {\dfrac {\partial {\vec {\xi }}}{\partial w}}=\det \left({\dfrac {\partial {\vec {\xi }}}{\partial u}}\,{\dfrac {\partial {\vec {\xi }}}{\partial v}}\,{\dfrac {\partial {\vec {\xi }}}{\partial w}}\right)=\det \left(J_{\vec {\xi }}\right)}$

The strung together partial gradients ( is a vector-valued function) precisely form the elements of the 3x3 Jacobi matrix . The associated Jacobian determinant, also known as the functional determinant, precisely calculates the additional factor for a coordinate transformation . ${\displaystyle {\vec {\xi }}}$

If the volume element is scalar, the factor is reduced to its Euclidean norm . After the volume integral has been parameterized, the integral can be calculated step by step with the help of Fubini 's theorem. ${\displaystyle \Vert {\vec {N}}\Vert }$

Special volume integrals with coordinate transformation

Cartesian and spherical coordinates

The conversion formulas for spherical coordinates result in the following substitution function ${\displaystyle (r,\theta ,\varphi )}$

${\displaystyle {\vec {\xi }}(r,\theta ,\varphi )={\begin{pmatrix}r\sin \theta \cos \varphi \\r\sin \theta \sin \varphi \\r\cos \theta \quad \end{pmatrix}}}$.

With the calculation of all partial gradients of the suitable Jacobi matrix is determined. Their determinant then results ${\displaystyle {\vec {\xi }}(r,\theta ,\varphi )}$

${\displaystyle {\vec {N}}=\det \left(J_{\vec {\xi }}\right)=r^{2}\sin \theta }$.

This results in spherical coordinates for the volume element

${\displaystyle \mathrm {d} V={\vec {N}}\,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \varphi .}$

The volume integral is then

${\displaystyle \iiint _{V_{Kart}}{\vec {f}}(x,y,z)\mathrm {d} x\mathrm {d} y\mathrm {d} z=\iiint _{V_{Kug}}{\vec {f}}\left({\vec {\xi }}(r,\theta ,\varphi )\right)\cdot r^{2}\sin \theta \,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \varphi .}$

The reverse transformation with the inverse function is also possible

${\displaystyle {\vec {\xi }}^{-1}(x,y,z)={\begin{pmatrix}{\sqrt {x^{2}+y^{2}+z^{2}}}\\{\frac {\pi }{2}}-\arctan {\frac {z}{\sqrt {x^{2}+y^{2}}}}\\\arctan \left({\frac {y}{x}}\right)\end{pmatrix}}}$.

With the calculation of all partial gradients of the suitable Jacobi matrix is ​​determined. Their determinant then results ${\displaystyle {\vec {\xi }}^{-1}(x,y,z)}$

${\displaystyle {\vec {N}}=\det \left(J_{{\vec {\xi }}^{-1}}\right)={\frac {1}{{\sqrt {x^{2}+y^{2}}}(x^{2}+y^{2}+z^{2})}}}$.

Cartesian to cylindrical coordinates

The conversion formulas for cylindrical coordinates result in the following substitution function ${\displaystyle (\rho ,\varphi ,z)}$

${\displaystyle {\vec {\xi }}(\rho ,\varphi ,z)={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\quad \end{pmatrix}}}$.

With the calculation of all partial gradients of the suitable Jacobi matrix is ​​determined. Their determinant then results ${\displaystyle {\vec {\xi }}(\rho ,\varphi ,z)}$

${\displaystyle {\vec {N}}=\det \left(J_{\vec {\xi }}\right)=\rho }$.

This results in cylindrical coordinates for the volume element

${\displaystyle \mathrm {d} V={\vec {N}}\,\mathrm {d} \rho \mathrm {d} \varphi \mathrm {d} z=\rho \,\mathrm {d} \rho \mathrm {d} \varphi \mathrm {d} z.}$

The volume integral is then

${\displaystyle \iiint _{V_{Kart}}{\vec {f}}(x,y,z)\mathrm {d} x\mathrm {d} y\mathrm {d} z=\iiint _{V_{Zyl}}{\vec {f}}\left({\vec {\xi }}(\rho ,\varphi ,z)\right)\cdot \rho \,\mathrm {d} \rho \mathrm {d} \varphi \mathrm {d} z.}$

application

Volume integrals are used in many physical problems. In this way, the respective underlying variables can be calculated from all densities in the case of volume integration, for example the electrical charge from the charge density or the mass from the (mass) density . The Gaussian integral theorem , which is particularly important in electrodynamics , is also based on a volume integral. The probability density of the speed value in the Maxwell-Boltzmann distribution results from volume integration over the distribution of the individual directions of the speed vector - this is an example of a volume integral over a non-geometric volume.

If the integrand is used as the function that is constantly equal to 1 on the integration volume, a formula for the volume measure is obtained

${\displaystyle \mathrm {vol} (V)=\iiint _{V}\mathrm {d} V}$.

Examples

Examples for dealing with volume integrals can be found here:

• Calculation of the moment of inertia of a homogeneous solid sphere
• Derivation of the velocity distribution of the Maxwell-Boltzmann distribution

Further information

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