In general relativity we demand that the physical law can be stated as a form which does not depend on the choose of particular coordinate system, So the vector field is defined as a changing object following a regular pattern under the transformation of coordinates. For example, we can define the covariant vector field as:(adsbygoogle = window.adsbygoogle || []).push({});

##v^\mu (x) \rightarrow v'^{\mu} (x)=J^\mu_\nu(x) v^\nu (x)## . ##J^\mu_\nu(x)\equiv \frac{\partial x'^{\mu}}{\partial x^\nu}##

In fact, this transformation is a global transformation. What interested me is the infinitesimal transformation induced by the global transformation between two neighboring point:##x## and ##x+dx##.

If we require that ##v(x)=G(x)v(x_0)##, or ##dv(x)=(\partial_iG(x))G^{-1}(x)dx^iv(x)##

Then ## v'(x)=J(x)G(x)v(x_0)##. So the infinitesimal change is:

## dv'(x)=\partial_i(J(x)G(x)) (J(x)G(x))^{-1}dx^iv'(x)##,

And we can get:

## dv'(x)=\left((\partial_iJ)J^{-1} +(\partial_i G)G^{-1} +J[(\partial_i G)G^{-1}, J^{-1}]\right)dx^i v'(x)##

To us, the first two terms ##(\partial_i J)J^{-1}dx^iv'(x)## and##(\partial_i G)G^{-1}dx^i v'(x)## is easily to be explained, They can be seemed to result from the original vector field and coordinate transformation, but the third term is hard to be explained, Does it has any exact physical interpretations?

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# I A question about the definition of vector field

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