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tensor[raise] - raise a covariant index
tensor[lower] - lower a contravariant index
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Calling Sequence
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raise(contravariant_metric_tensor, A, i1, i2, ... )
lower(covariant_metric_tensor, A, i1, i2, ... )
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Parameters
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contravariant_metric_tensor
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metric tensors used to raise the indices
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covariant_metric_tensor
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metric tensors used to lower the indices
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A
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tensor by which to raise/lower the indices
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i1, ...
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non-empty sequence of indices of A to raise/lower
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Description
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The function raise(con_met, A, 2, 3) computes the tensor A with indices 2 and 3 raised using the contravariant metric con_met.
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The function lower(cov_met, A, 1, 4) computes the tensor A with indices 1 and 4 lowered using the covariant metric cov_met.
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Each index in the call to raise must be a valid covariant index of A.
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Each index in the call to lower must be a valid contravariant index of A.
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There must be at least 1 index given and the number of indices cannot exceed the rank of A.
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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.
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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](..).
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Examples
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>
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covariant Euclidean 3-space metric in spherical-polar coordinates:
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![a := create([-1, -1], array([[1, 0, 0], [0, r^2, 0], [0, 0, r^2*sin(theta)^2]]))](/support/helpjp/helpview.aspx?si=5528/file04543/math152.png)
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![a := TABLE([index_char = [-1, -1], compts = matrix([[1, 0, 0], [0, r^2, 0], [0, 0, r^2*sin(theta)^2]])])](/support/helpjp/helpview.aspx?si=5528/file04543/math155.png)
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contravariant Euclidean 3-space metric in spherical-polar coordinates:
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![A := create([1, 1], array([[1, 0, 0], [0, 1/r^2, 0], [0, 0, 1/(r^2*sin(theta)^2)]]))](/support/helpjp/helpview.aspx?si=5528/file04543/math161.png)
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![A := TABLE([index_char = [1, 1], compts = matrix([[1, 0, 0], [0, 1/r^2, 0], [0, 0, 1/(r^2*sin(theta)^2)]])])](/support/helpjp/helpview.aspx?si=5528/file04543/math164.png)
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create a mixed 2-tensor, raise one index, then lower the other
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![T := create([1, -1], array([[w, x, 0], [y, z, 0], [0, y^2, x*y*w]]))](/support/helpjp/helpview.aspx?si=5528/file04543/math170.png)
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![T := TABLE([index_char = [1, -1], compts = matrix([[w, x, 0], [y, z, 0], [0, y^2, x*y*w]])])](/support/helpjp/helpview.aspx?si=5528/file04543/math173.png)
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![TABLE([index_char = [1, 1], compts = matrix([[w, x/r^2, 0], [y, z/r^2, 0], [0, y^2/r^2, x*y*w/(r^2*sin(theta)^2)]])])](/support/helpjp/helpview.aspx?si=5528/file04543/math180.png)
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![TABLE([index_char = [-1, -1], compts = matrix([[w, x, 0], [r^2*y, r^2*z, 0], [0, r^2*sin(theta)^2*y^2, r^2*sin(theta)^2*x*y*w]])])](/support/helpjp/helpview.aspx?si=5528/file04543/math187.png)
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