Covariant differentiation exercise 1: calculation in cylindrical coordinates

To make the meaning of the equations of covariant differentiation seen in last article Introduction to Covariant Differentiation more explicit, we will consider the covariant derivative of vector V with respect  to θ in cylindrical coordinates (so x¹=r, x²=θ, and x³=z).

Setting β=2 in the following equation, since we are interested in the covariant derivative with respect to θ:

we get

We know the values of the first two Christoffel symbol as we have already calcuted them in the previous article Christoffel symbol exercise: calculation in polar coordinates part I

so that we already  know that

We know also that since

all the symbols from the following form vanish

thus we end up with this equality