Parallel transport, the solution

Posted by on Jan 27, 2013 in Science | 0 comments

So far, in previous posts, I’ve discussed how spaces with an irreducible curvature yield a failure in parallel transport around a closed path, how the usual derivative of a vector fails to yield another proper vector, and how the covariant derivative solves this problem. In this post, I want to show how the covariant derivative can solve the problem of parallel transport along a path. The idea is the essence of simplicity. Suppose you have a vector, . You want to transport this along some arbitrary curve in some arbitrary manifold of some arbitrary dimension. Let a differential element...

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Let’s differentiate tensors!

Posted by on Jan 20, 2013 in Science | 0 comments

This is a nasty and highly mathematical topic, but I am going to try to keep this presentation of it as absolutely simple as possible. The question is, can we differentiate tensors and get tensors? Jumping ahead, the answer is no, not unless the process is done in a very special way. This will take us to the notion of covariant derivates and the Christoffel symbol, but I’ll go in simple steps. First, imagine that we have a scalar field defined as a constant everywhere. This could be a uniform temperature in some region of space; maybe a uniform humidity or electric potential. It really...

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