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Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.

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Derivative chain rule in a triangle, confusing but interesting problem

It may be a mistake in Eq.4. Your function $T(r,z)$ does not depend on $\xi$ and has nothing in principle to do with triangles. Since you are only interested in values of $T$ for $(r,z)$ pairs that …
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