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Integration in higher dimensions

  • M. Schreiber
Part of the Universitext book series (UTX)

Abstract

Let P be a rectangular parallelopiped in ℝk, with edges along the Cartesian coordinate axes. We call such a figure a Cartesian rectangle. The figure shows a Cartesian rectangle in ℝ2. Let the edges of P have lengths dx1,dx2,...,dxk. 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)
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
$$ \frac{{{\rm{Vol(T(P))}}}}{{{\rm{Vol(P)}}}}{\rm{ }} = {\rm{ |T| ,}} $$
(2)
where |T| is the standard notation for the determinant of T.(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

Authors and Affiliations

  • M. Schreiber
    • 1
  1. 1.Department of MathematicsThe Rockefeller UniversityNew YorkUSA

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