Abstract
In a (pseudo-)Riemannian space χ, we can measure not only lengths but also volumes, i.e., we can define a natural volume density dV on χ. Indeed, let (U, h) be an arbitrary chart of an arbitrary (pseudo-)Riemannian space χ, let ||g ij || be the matrix of components of the metric tensor g in the chart (U, h), and let
be its determinant. The transformation formula for the matrix of a quadratic form under a change of basis directly implies that under a change of coordinates, the determinant det g is multiplied by the square of the Jacobian of the transition functions. Therefore, setting
(as usual, we mean the arithmetical square root), we obtain a certain volume density dV on χ. (Of course, for a Riemannian space, it is not necessary to pass from det g to | det g|.) The density dV is conventionally denoted by
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© 2001 Springer-Verlag Berlin Heidelberg
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Postnikov, M.M. (2001). Harmonic Functionals and Related Topics. In: Geometry VI. Encyclopaedia of Mathematical Sciences, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04433-9_13
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DOI: https://doi.org/10.1007/978-3-662-04433-9_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07434-9
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