find the inverse of a metric tensor - Maple Help

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Tensor[InverseMetric] - find the inverse of a metric tensor

Calling Sequences

     InverseMetric(g)

Parameters

   g    - a metric tensor

Description

• 

A metric tensor g is a symmetric, non-degenerate, rank 2 covariant tensor. The inverse of a metric tensor is a symmetric, non-degenerate, rank 2 contravariant tensor g. The components of h are given by the inverse of the matrix defined by the components of g.

• 

InverseMetric(g) calculates the inverse of the metric tensor g.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form InverseMetric(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-InverseMetric.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a manifold M and define a metric tensor g on the tangent space of M.

DGsetupx,y,M

frame name: M

(2.1)
M > 

g:=evalDGxdx &t dxdy &t dy

gxdxdxdydy

(2.2)

 

Calculate the inverse of g.

M > 

h:=InverseMetricg

h1xD_xD_xD_yD_y

(2.3)

 

Check the result -- the contraction of h with g should be the type (1, 1) tensor whose components are the identity matrix.

M > 

ContractIndicesg,h,1,1

dxD_x+dyD_y

(2.4)

 

Example 2.

First create a rank 3 vector bundle EM and define a metric gon the fibers.

M > 

DGsetupx,y,u,v,w,E

frame name: E

(2.5)
E > 

g:=evalDGdu &t dudv &t dwdw &t dv

gdududvdwdwdv

(2.6)

 

Calculate the inverse of g.

E > 

InverseMetricg

D_uD_uD_vD_wD_wD_v

(2.7)

See Also

DifferentialGeometry, Tensor, ContractIndices, RaiseLowerIndices, Physics[g_]


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