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tensor

 raise
 raise a covariant index
 lower
 lower a contravariant index

 Calling Sequence raise(contravariant_metric_tensor, A, i1, i2, ... ) lower(covariant_metric_tensor, A, i1, i2, ... )

Parameters

 contravariant_metric_tensor - metric tensors used to raise the indices covariant_metric_tensor - metric tensors used to lower the indices A - tensor by which to raise/lower the indices i1, ... - non-empty sequence of indices of A to raise/lower

Description

Important: The tensor package has been deprecated. Use the superseding command DifferentialGeometry[Tensor][RaiseLowerIndices].

 • The function raise(con_met, A, 2, 3) computes the tensor A with indices 2 and 3 raised using the contravariant metric con_met.
 • The function lower(cov_met, A, 1, 4) computes the tensor A with indices 1 and 4 lowered using the covariant metric cov_met.
 • Each index in the call to raise must be a valid covariant index of A.
 • Each index in the call to lower must be a valid contravariant index of A.
 • There must be at least 1 index given and the number of indices cannot exceed the rank of A.
 • Simplification:  These routines use the tensor/prod/simp routine for simplification purposes.  The simplification routine is applied to each component of the result after it is computed.  By default, tensor/prod/simp is initialized to the tensor/simp routine. It is recommended that the tensor/prod/simp routine be customized to suit the needs of the particular problem.
 • These functions are part of the tensor package, and so can be used in the form raise(..) / lower(..) only after performing the command with(tensor), or with(tensor, raise) / with(tensor, lower).  These functions can always be accessed in the long form tensor[raise](..) / tensor[lower](..).

Examples

Important: The tensor package has been deprecated. Use the superseding command DifferentialGeometry[Tensor][RaiseLowerIndices].

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

covariant Euclidean 3-space metric in spherical-polar coordinates:

 > $a≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{array}\left(\left[\left[1,0,0\right],\left[0,{r}^{2},0\right],\left[0,0,{r}^{2}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}\right]\right]\right)\right)$
 ${a}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& {{r}}^{{2}}& {0}\\ {0}& {0}& {{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (1)

contravariant Euclidean 3-space metric in spherical-polar coordinates:

 > $A≔\mathrm{create}\left(\left[1,1\right],\mathrm{array}\left(\left[\left[1,0,0\right],\left[0,\frac{1}{{r}^{2}},0\right],\left[0,0,\frac{1}{{r}^{2}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}}\right]\right]\right)\right)$
 ${A}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{1}& {0}& {0}\\ {0}& \frac{{1}}{{{r}}^{{2}}}& {0}\\ {0}& {0}& \frac{{1}}{{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{1}\right]\right]\right)$ (2)

create a mixed 2-tensor, raise one index, then lower the other

 > $T≔\mathrm{create}\left(\left[1,-1\right],\mathrm{array}\left(\left[\left[w,x,0\right],\left[y,z,0\right],\left[0,{y}^{2},xyw\right]\right]\right)\right)$
 ${T}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{w}& {x}& {0}\\ {y}& {z}& {0}\\ {0}& {{y}}^{{2}}& {x}{}{y}{}{w}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{-1}\right]\right]\right)$ (3)
 > $\mathrm{raise}\left(A,T,2\right)$
 ${table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{w}& \frac{{x}}{{{r}}^{{2}}}& {0}\\ {y}& \frac{{z}}{{{r}}^{{2}}}& {0}\\ {0}& \frac{{{y}}^{{2}}}{{{r}}^{{2}}}& \frac{{x}{}{y}{}{w}{}{{\mathrm{csc}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{{{r}}^{{2}}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{1}\right]\right]\right)$ (4)
 > $\mathrm{lower}\left(a,T,1\right)$
 ${table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{w}& {x}& {0}\\ {{r}}^{{2}}{}{y}& {{r}}^{{2}}{}{z}& {0}\\ {0}& {{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{y}}^{{2}}& {{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{x}{}{y}{}{w}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (5)