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Tensor[TensorInnerProduct] - compute the inner product of two vectors, forms or tensors with respect to a given metric tensor
Calling Sequences
TensorInnerProduct(g, T, S)
Parameters
g - a covariant metric tensor on a manifold M
T, S - two vector fields, forms or tensors (with the same index type) on M
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Description
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Let P = TensorInnerProduct(g, T, S). Let g = g_{ij} dx^i &t dx^j and let h = h_{ij} D_x^i &t D_x^j be the inverse contravariant metric. If T = t^i*D_x^i and S = s^i*D_x^i are vectors, then P = g_{ij}*t^i*s^j. If T = t_i*dx^i and S = s_i*dx^i are 1-forms, then P = h^{ij} t_i*s_j. If T= t_{ij}*dx^i &w dx^j and S= s_{ij}*dx^i &w dx^j are 2-forms, then P = h^{ik}*h^{jl}*t_{ij}*s_{kl}. If T = t^i_{jk}*D_x^i &t dx^j &t dx^k and S = s^i_{jk}*D_x^i &t dx^j &t dx^k are type [1, 2] tensors, then P = g_{ia}h^{jb}*h^{kc}*t_^i{jk}*s^a{bc} and so on.
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form TensorInnerProduct(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-TensorInnerProduct.
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Examples
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First define a manifold M with local coordinates [x, y] and define an (covariant) metric on M.
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![DGsetup([x, y, z], M)](/support/helpjp/helpview.aspx?si=5621/file05929/math88.png)
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Example 1.
Compute the inner product of two vectors.
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Example 2.
Compute the inner product of two 1-forms.
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Example 3.
Compute the inner product of two 2-forms.
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Example 4.
Compute the inner product of two 2-tensors.
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| (2.13) |
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