Differential Forms pp 37-73 | Cite as

# Integration in higher dimensions

Chapter

## Abstract

Let P be a rectangular parallelopiped in ℝ
the product of its edge lengths. Let T be a k-by-k matrix. We may apply T to each vector lying in P and in this way transform P to another figure T(P), which typically will no longer be a Cartesian rectangle, nor even rectangular. To calculate the volume Vol(T(P)) of T(P) we use the theorem on the geometric meaning of the determinant which states, in the present notation, that
where |T| is the standard notation for the determinant of T.

^{k}, with edges along the Cartesian coordinate axes. We call such a figure a Cartesian rectangle. The figure shows a Cartesian rectangle in ℝ^{2}. Let the edges of P have lengths dx_{1},dx_{2},...,dx_{k}. then its k-dimensional volume, denoted Vol(P), is be definition$$
{\rm{Vol(P) = d}}{{\rm{x}}_{\rm{1}}}{\rm{ d}}{{\rm{x}}_{\rm{2}}}{\rm{ }}...{\rm{ d}}{{\rm{x}}_{\rm{k}}}{\rm{ ,}}
$$

(1)

$$
\frac{{{\rm{Vol(T(P))}}}}{{{\rm{Vol(P)}}}}{\rm{ }} = {\rm{ |T| ,}}
$$

(2)

^{(1)}Putting this together with (1) we have$$
{\rm{Vol(T(P)) = |T|d}}{{\rm{x}}_{\rm{1}}}{\rm{d}}{{\rm{x}}_{\rm{2}}}...{\rm{d}}{{\rm{x}}_{\rm{k}}}.
$$

(3)

## Keywords

Volume Element Tangent Space Implicit Function Theorem Maximal Rank Rectangular Parallelopiped
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag, New York Inc. 1977