Description Usage Arguments Details Author(s) References See Also Examples

The volume element in *n* dimensions

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`n` |
Dimension of the space |

`K` |
Object of class |

Spivak phrases it well (theorem 4.6, page 82):

If *V* has dimension *n*, it follows that *.*
has dimension 1. Thus all alternating *n*-tensors on *V* are
multiples of any non-zero one. Since the determinant is an example of
such a member of *.* it is not surprising to find it
in the following theorem:

Let *.* be a basis for *V* and let
*.*. If *.*
then

*ommitted; see PDF*

(see the examples for numerical verification of this).

Neither the zero *k*-form, nor scalars, are considered to be a
volume element.

Robin K. S. Hankin

Spivak

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