Functional determinant

The functional determinant or Jacobi determinant is a mathematical variable that plays a role in the multi-dimensional integral calculus , i.e. the calculation of surface and volume integrals. In particular, it is used in the area formula and the transformation set resulting from it .

Local behavior of a function

The functional determinant gives important information about the behavior of the function in the vicinity of this point at a given point. If, for example, the functional determinant of a continuously differentiable function is not equal to zero at a point , then the function is invertible in a neighborhood of . Furthermore, if there is a positive determinant in the function, its orientation is retained and if the functional determinant is negative, the orientation is reversed. The absolute value of the determinant in the point indicates the value at which the function expands or shrinks near . ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

definition

For a differentiable function , the functional determinant is defined as the determinant of the Jacobi matrix of , i.e. as ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle f}$

${\displaystyle \det \,Df(x)}$

With

${\displaystyle Df(x)=\left({\frac {\partial f_{i}}{\partial x_{j}}}(x)\right)_{i,j=1,\dotsc ,n}}$.

This definition is sufficient for the transformation of volume elements, an important application in physics. The area formula of the theory of measure and integration, on the other hand, also describes how integrals are transformed via functions that map spaces of different dimensions into one another. In this use case, there is no longer a square matrix, so the expression above is no longer defined. The following definition is then used: ${\displaystyle Df}$

The generalized functional determinant of a function is defined as ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$

${\displaystyle {\mathcal {J}}\!f(x):={\sqrt {\det \left(\left(Df(x)\right)^{T}\cdot Df(x)\right)}}}$

The Jacobi matrix and its transpose denotes . The term is called the Gram determinant of . ${\displaystyle Df(x)\in \mathbb {R} ^{m\times n}}$ ${\displaystyle (Df(x))^{T}}$ ${\displaystyle \det \left(\left(Df(x)\right)^{T}\cdot Df(x)\right)}$ ${\displaystyle Df}$

As long as the image under consideration is not a self- image , it is customary to omit the prefix generalized . In the case of self-portrayals, however, this can lead to misunderstandings, since the two definitions generally assume different values. It is true

${\displaystyle {\mathcal {J}}\!f(x)={\sqrt {(\det Df)^{2}}}=|\det Df|\neq \det Df}$

In the context of the area or transformation formula, the amount is always used anyway.

Examples

When integrating using geometric objects, it is often impractical to integrate using Cartesian coordinates . In physics, for example, the integral over a radially symmetrical potential field , the value of which only depends on a radius , can be calculated much more easily in spherical coordinates . ${\displaystyle r}$

To do this, one applies a coordinate transformation . According to the transformation theorem, the following applies in this example: ${\displaystyle \Phi }$

${\displaystyle \int _{\Omega }U({\vec {r}})dV=\int _{\Phi ^{-1}(\Omega )}U(\Phi (r,\theta ,\varphi ))\cdot \left|\det D\Phi (r,\theta ,\varphi )\right|\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi }$

The following are calculations for three coordinate systems:

Polar coordinates

The formulas for converting polar coordinates to Cartesian coordinates are:

${\displaystyle x=r\cos \varphi }$

${\displaystyle y=r\sin \varphi }$

The functional determinant is thus:

${\displaystyle \det {\frac {\partial (x,y)}{\partial (r,\varphi )}}=\det {\begin{pmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{pmatrix}}=\det {\begin{pmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{pmatrix}}=r\cdot (\cos \varphi )^{2}+r\cdot (\sin \varphi )^{2}=r}$

The following results for the surface element : ${\displaystyle \mathrm {d} A}$

${\displaystyle \mathrm {d} A=\left|\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}\right|\,\mathrm {d} r\,\mathrm {d} \varphi =r\,\mathrm {d} r\,\mathrm {d} \varphi .}$

Spherical coordinates

The conversion formulas from spherical coordinates ( ) to Cartesian coordinates are: ${\displaystyle r,\theta ,\varphi }$

${\displaystyle x=r\sin \theta \cos \varphi }$,

${\displaystyle y=r\sin \theta \sin \varphi }$ and

${\displaystyle z=r\cos \theta \quad }$.

The functional determinant is thus:

${\displaystyle \det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}=\det {\begin{pmatrix}\sin \theta \cos \varphi &r\cos \theta \cos \varphi &-r\sin \theta \sin \varphi \\\sin \theta \sin \varphi &r\cos \theta \sin \varphi &r\sin \theta \cos \varphi \\\cos \theta &-r\sin \theta &0\end{pmatrix}}=r^{2}\sin \theta .}$

Hence the following results for the volume element : ${\displaystyle \mathrm {d} V}$

${\displaystyle \mathrm {d} V=\left|\det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi .}$

Sometimes it is more practical to use the following convention:

${\displaystyle x=r\cos \theta \cos \varphi }$,

${\displaystyle y=r\cos \theta \sin \varphi }$ and

${\displaystyle z=r\sin \theta \quad }$.

The functional determinant is thus:

${\displaystyle \det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}=\det {\begin{pmatrix}\cos \theta \cos \varphi &-r\sin \theta \cos \varphi &-r\cos \theta \sin \varphi \\\cos \theta \sin \varphi &-r\sin \theta \sin \varphi &r\cos \theta \cos \varphi \\\sin \theta &r\cos \theta &0\end{pmatrix}}=-r^{2}\cos \theta .}$

So we get for the volume element : ${\displaystyle \mathrm {d} V}$

${\displaystyle \mathrm {d} V=\left|\det {\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\cos \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi .}$

Cylindrical coordinates

The conversion formulas of cylindrical coordinates ( , , ) are denominated in Cartesian coordinates: ${\displaystyle \rho }$ ${\displaystyle \varphi }$ ${\displaystyle z}$

${\displaystyle {\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\,.\end{aligned}}}$

The functional determinant is thus:

${\displaystyle \det {\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}=\det {\begin{pmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{pmatrix}}=\rho .}$

Hence the following results for the volume element : ${\displaystyle \mathrm {d} V}$

${\displaystyle \mathrm {d} V=\left|\det {\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}\right|\,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z.}$

A different sequence of the cylinder coordinates could just as well have been chosen. The functional determinant then reads, for example:

${\displaystyle \det {\frac {\partial (x,y,z)}{\partial (\rho ,z,\varphi )}}=\det {\begin{pmatrix}\cos \varphi &0&-\rho \sin \varphi \\\sin \varphi &0&\rho \cos \varphi \\0&1&0\end{pmatrix}}=-\rho .}$

However, only the amount of the determinant is included in the transformation law , so the result is then independent of the selected order of the variables according to which the derivation is made.

literature

• Herbert Federer : Geometric measure theory . 1st edition. Springer, Berlin 1996, ISBN 3-540-60656-4 (English). (For the definition)
• Wolfgang Nolting : Classic mechanics . In: Basic course in theoretical physics . 8th edition. tape 1 . Springer, Berlin 2006, ISBN 978-3-540-34832-0 . (For the examples and the special case of ) ${\displaystyle \mathbb {R} ^{3}}$