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 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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