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Vector field proof

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Please solve attached part b.

Suppose that x<-->A(x) is a smooth mapping from a neighborhood U of R^n containing 0 into the SO(n) matrices, defined by the relations....Show that for any vector field X on U.....

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Solution Summary

This provides an example of completing a proof regarding a vector field and a smooth mapping.

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In terms of the entries, matrix multiplication is given by
Important note: here and in what follows I am using the Einstein summation convention, in which we sum over all possible values of any repeated indices . So the right hand side of the above is really shorthand for the sum
Since transposing just involves interchanging rows and columns, we see that
Combining these two facts, we get:
Everything so far has been ...

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