PHYMAT.PRO - Diego E. Gallardo - Cranfield University

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THE METRIC TENSOR

The Metric Tensor g: definition

The metric tensor g=(g_ij) is defined as a matrix field that satisfies the following points in every admissible coordinate system:

g is class C²
g is symetric (g_ij=g_ji)
g is no-singular (|g_ij|!=0)
The differential form of distance ds² is invariant with coordinate changes

Arc's length parameter:

The arc's length parameter of a curve xⁱ(t) (a≤t≤b) is defined as:

(1.1)

with e=+1 or -1 (metric indicator)
Deriving (1.1) and squaring:

(1.2)

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Metric tensor for some coordinate systems

Cylindric:

Spherical:

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