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Once a local coordinate system xi is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation gij is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein notation for implicit sums.
The length of a segment of a curve parameterized by t, from a to b, is defined as:
- G = JTJ
The Euclidean metric
Given a two-dimensional Euclidean metric tensor:
The length of a curve reduces to the familiar calculus formula:
The Euclidean metric in some other common coordinate systems can be written as follows.
Polar coordinates: (x1,x2) = (r,θ)
Cylindrical coordinates: (x1,x2,x3) = (r,θ,z)
Spherical coordinates: (x1,x2,x3) = (r,φ,θ)
Flat Minkowski space: (x1,x2,x3,x4) = (t,x,y,z)
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