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Example 1.
First create a 2 dimensional manifold and define a connection on the tangent space of .
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M >
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| (2.2) |
Define some vector fields and tensor fields and compute the directional covariant derivative with respect to .
M >
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M >
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M >
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M >
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| (2.7) |
M >
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M >
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M >
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| (2.9) |
Example 2.
Define a frame on and use this frame to specify a connection on the tangent space of .
M >
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M1 >
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| (2.11) |
Define a vector field and a tensor field and compute the directional covariant derivative with respect to .
M1 >
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M1 >
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| (2.12) |
M1 >
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| (2.13) |
Example 3.
First create a rank 3 vector bundle and define a connection on .
M1 >
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E >
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| (2.15) |
E >
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E >
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| (2.17) |
To covariantly differentiate a mixed tensor on , a connection on is also needed.
E >
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| (2.19) |
E >
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E >
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| (2.21) |