Area formula

The area formula is understood to be an integration rule for the calculation of areas of -dimensional areas in ( ). Here, it is assumed that the dimensional surface is parameterized, ie, there is one in an area defined injective differentiable map and a measurable subset so that the image of under the mapping is . ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m\leq n}$ ${\displaystyle m}$ ${\displaystyle A\subset \mathbb {R} ^{n}}$ ${\displaystyle G\subset \mathbb {R} ^{m}}$ ${\displaystyle f\colon G\to \mathbb {R} ^{n}}$ ${\displaystyle B\subset G}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle A=f(B)}$

Then:

${\displaystyle {\mathcal {H}}^{m}(A)=\int \limits _{B}{\sqrt {\det(Df^{T}\,Df)}}\,dL^{m},}$

It is the -dimensional Hausdorff measure (the dimensional surface area) of , and the dimensional Lebesgue measure (of volume) in the . The integrand is called the generalized Jacobian determinant of ; is the derivative ( functional matrix ) of and its transpose . ${\displaystyle {\mathcal {H}}^{m}(A)}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle A}$ ${\displaystyle L^{m}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle f}$ ${\displaystyle Df}$ ${\displaystyle f}$ ${\displaystyle Df^{T}}$

A more general formulation of the area formula is

${\displaystyle \int \limits _{A}\!g\,\mathrm {d} {\mathcal {H}}^{m}=\int \limits _{B}\!(g\circ f){\sqrt {\det(Df^{T}\,Df)}}\,\mathrm {d} L^{m}}$

and delivers the value of the integral of a function defined on the surface according to the Hausdorff measure . ${\displaystyle A}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {H}}^{m}}$

As conditions for these formulas are - measurability of and -Messbarkeit of call, but this does not mean a significant limitation, since all have this property in practice occurring levels or functions. ${\displaystyle L^{m}}$ ${\displaystyle B}$ ${\displaystyle {\mathcal {H}}^{m}}$ ${\displaystyle g}$

Both formulas only apply in this form if the mapping (except for a zero set) is injective, because on the left side each pixel is calculated only once, on the right, however, each original pixel. ${\displaystyle f}$

The requirement that the function is differentiable can be weakened. It is enough if it is Lipschitz continuous ; then it can automatically be differentiated almost everywhere. ${\displaystyle f}$

In the special case , the area formula results in the transformation formula from the theory of dimensions and integration. ${\displaystyle m=n}$

literature

• Herbert Federer : Geometric measure theory . 1st edition. Springer, Berlin 1996, ISBN 3-540-60656-4 (English).