Potential Energy
Theorem
Let $M$ be a compact surface, and let $V$ be a vector field on $M$ with finitely many zeros. Then the following holds.
$$ I(M) = \chi(M) $$
Here $\chi$ denotes the Euler characteristic.
Explanation
This is called the Poincaré–Brouwer theorem.
It’s quite remarkable, although the statement appears straightforward once one examines the proof.
Proof
$$ I(W) = \sum i_{p} = (+1)V + (-1)E + (+1)F = \chi(M) $$
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