## A tensor crib sheet

Here are a few useful formulae for work with tensors: aka the Christoffel symbol is not a...

Read More## Let’s differentiate tensors!

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...

Read More## The metric tensor

The name, metric tensor, is enough to strike fear into the hearts of any physics student, associated as it is with General Relativity. I’m here to tell you that, in its heart of hearts, the metric tensor is not much more complex than the standard vector dot product, of which it is a generalization. In what follows, I hope to remove some of the tension surrounding the metric tensor. That raises a pretty basic question right off the bat: if we already have the machinery of the dot product, why do we need a metric tensor? Where does the dot product break down? Recall that one way to...

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