- #1

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- A
- Thread starter mertcan
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- #1

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- #2

Tio Barnabe

If you want to see the change of functions, you just use the ordinary derivative.

- #3

jedishrfu

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https://en.m.wikipedia.org/wiki/Covariant_derivative

It can give you some insight on its usage and purpose.

- #4

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If you do not, the result will not be invariant.but my question is why do we have to use covariant derivative only with tensors? ?? Is there a logic of this situation? ???

Since you labeled this thread "A": A priori, a tensor field is a section over the corresponding tensor bundle. Since the tensors at different points of the base manifold belong to different tensor spaces. As you will remember from basic calculus, derivatives were defined by taking differences of functions at different points and studying how it behaves as those points approach each other. However, for tensor fields there is a priori no natural way of taking differences of tensors at different points of the manifold and in order to be able to define a derivative we therefore must introduce a connection that can be used to relate the tensor spaces at different points to each other. This connection defines the covariant derivative and it makes no sense to talk about the "ordinary" derivative because it is unclear what it would mean.

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