The
Metric Tensor
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Definition of Metric Tensor [1]: A multilinear map from vectors to the set of real numbers which is used for calculating the magnitude of a vector of the inner product of two vectors, which is symmetric, is called the metric tensor and represented here as g. This definition can be described in part as
The
completion of this definition is in the multilinearity, which is a property all
tensors have by definition. The magnitude of a vector in terms of Eq. (1) is
given by
The components
of the metric tensor are defined as
A common
use of the metric tensor is to define a distance function on a manifold. The
interval ds2
between two points is defined as
The distance between the two points separated by the infinitesimal displacement dX is the square root of the interval. In Most Euclidean applications the interval is defined as the square root of dX·dX.
References
[1] Geometrical methods of mathematical physics, by Bernard F. Schutz, page 64.