Integration in higher dimensions

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


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
$$ {\rm{Vol(P) = d}}{{\rm{x}}_{\rm{1}}}{\rm{ d}}{{\rm{x}}_{\rm{2}}}{\rm{ }}...{\rm{ d}}{{\rm{x}}_{\rm{k}}}{\rm{ ,}} $$
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
$$ \frac{{{\rm{Vol(T(P))}}}}{{{\rm{Vol(P)}}}}{\rm{ }} = {\rm{ |T| ,}} $$
where |T| is the standard notation for the determinant of T.(1) Putting this together with (1) we have
$$ {\rm{Vol(T(P)) = |T|d}}{{\rm{x}}_{\rm{1}}}{\rm{d}}{{\rm{x}}_{\rm{2}}}...{\rm{d}}{{\rm{x}}_{\rm{k}}}. $$


Volume Element Tangent Space Implicit Function Theorem Maximal Rank Rectangular Parallelopiped 
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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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