Tensor[CovariantDerivative] - calculate the covariant derivative of a tensor field with respect to a connection
Calling Sequences
CovariantDerivative(T, C1, C2)
Parameters
T - a tensor field
C1 - a connection
C2 - (optional) a second connection, needed when the tensor T is a mixed tensor defined on a vector bundle E -> M
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Description
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Let M be a manifold, let nabla be a linear connection on the tangent bundle of M, and let T be a tensor field on M. Then the covariant derivative of T with respect to nabla is nabla(T) = nabla_{E_i}(T) &t theta^i, where the vector fields E_1, E_2, ..., E_n define a local frame on M with dual coframe theta_1, theta_2, ..., theta_n. The tensor nabla_{E_i}(T) is the directional covariant derivative of T with respect to nabla in the direction of E_i. The definition of the covariant derivative for sections of a vector bundle E -> M and for mixed tensors on E is similar.
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CovariantDerivative(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-CovariantDerivative.
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Examples
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Example 1.
First create a 2 dimensional manifold M and define a connection on the tangent space of M.
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M >
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Define some tensor fields and compute their covariant derivatives with respect to C1.
M >
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M >
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M >
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M >
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M >
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M >
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To obtain a directional covariant derivative in the direction of a vector field X from the covariant derivative, contract the last index of the covariant derivative against the vector field.
M >
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M >
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M >
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Example 2.
Define a frame on M and use this frame to specify a connection on the tangent space of M.
M >
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M >
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M1 >
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Define some tensor fields and compute their covariant derivatives with respect to C2.
M1 >
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M1 >
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M1 >
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M1 >
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Example 3.
First create a rank 3 vector bundle E on M and define a connection on E.
M1 >
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E >
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E >
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E >
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To covariantly differentiate a mixed tensor on E, a connection on M is also needed.
E >
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E >
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E >
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