Covariant derivatives are hard

This last weekend I had a super good time with a friend, just spending a few hours figuring out some math from the physics behemoth “Gravity” by Misner, Wheeler and Thorne. I have some experience with general relativity and differential geometry, but I’ve never made it very far into the field, so we just opened the textbook, somewhere in Chapter 2, and looked in for entertainment:

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What the hell is a tensor anyways?

As I’m studying more and more differential geometry and general relativity (and being the noob that I am), I have to go back to this definition over and over again, so I thought that writing it down for good can help me (and hopefully others) get a quick refresher into what is a Tensor anyways.

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