How can I prove that the differential operator with respect to covariant coordinates behaves contravariantly?

Solution 1:

When we have as usual $x_i=g_{ij}x^j$ then one of the definitions $$ \partial_i=\frac{\partial}{\partial x^i}\,,\quad\partial^i=\frac{\partial}{\partial x_i}\, $$ must be sacrificed so that the more standard relationship $$ \partial_i=g_{ij}\,\partial^j $$ holds. (I prefer to write $\partial_i=\frac{\partial}{\partial x_i}$ but that's only a minor convention.) In this Q&A it is shown using the chain rule that

• $x^j$ are components of a contravariant vector;
• the derivative $\partial_i$ is a covariant vector;
• a contravariant derivative does not exist (your Professor's claim?);
• $\partial^j$ should better be called derivative with an upper index;

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