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

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