10. Covariant Derivatives

Let us consider the two coordinate systems, $(u_{1},u_{2},...u_{n})$ and $({u}’_{1},{u}’_{2},...{u}’_{n})$. We can ask ourselves a question if the transformation of the derivative of the vector in $(x)$ coordinate system and the derivative of the vector in $(u)$ coordinate system transforms as a tensor. i.e.

                                                                                                      $(1)$

We can show that in general curvilinear system, the derivative of the vector does not transform as a tensor. There is another quantity called the covariant derivatives which do transforms as a tensor. That is to say that its form remains the same even when we change the coordinate system.

Now let us consider the derivative of, $\vec{V}=V^{\alpha }\vec{e}_{\alpha }$

                                                         

                                   

Now from previous note, we can reduce the above equation $(1)$ as follows,

                                             

                                          

The derivative of a vector must be a tensor, so $\left ( \frac{\partial V^{\alpha }}{\partial u_{\beta }} + \Gamma_{\beta \gamma }^{\alpha } V^{\gamma }\right )$ are the components of a tensor, called the covariant derivative. Conventionally we write the above derivative in the following notation,

                                                              

Or in the semi-colon notation as,

                                                        

Important note to point out in this expression is that, $\left ( \partial_{\beta } V^{\alpha}\right )$ comes from the change of components with position, while $\Gamma_{\beta \gamma }^{\alpha } V^{\gamma }$ comes from the change from the change of basis vectors with position.