Ordinary and covaraint derivative

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Hello
what is the meaning of covaraint derivative ?
where the ordinary derivative of a function whit respect to a variable is zero, it means that function doesn't depend on that variable.but what about covaraint derivative ?
for example the metric tensor may depends on coordinate but its covaraint derivative is zero.
 
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The classical (directional) derivative is dependent on your choice of coordinates. The covariant derivative is not; is it covariant in the sense that it is defined in such a way as to be independent of its description in a particular coordinate system. That is why it is written as a directional derivative plus a term that compensates for any changes in coordinates, expressed through the Christoffel Symbols. It is basically a generalization of the classical derivative into Riemann geometry.
 
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