Einstein Relatively Easy

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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

 

which says that a change in the r-component of vector V caused by a change in θ is caused both by a change in Vr with respect to θ and by a change in the basis vectors which causes a portion of V that was originally in the θ-direction to now point in the r-direction.

Likewise, for the change in Vθ as the value of θ is changed, we have

which says that a change in the θ-component of vector V caused by a change in θ is caused both by a change in Vθ with respect to θ and by a change in the basis vectors which causes a portion of V that was originally in the r-direction to now point in the  θ-direction.

 Thus finally the covariant derivative of vector V with respect to θ in cylindrical coordinates is

 

 

 

 

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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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