A long exact sequence of cohomology groups (zig-zag and diagram-chasing)

Exterior differentiation(This section is intended to introduce the background. Feel free to skip if you already know exterior differentiation.)There are several useful tools for vector calculus on $\mathbb{R}^3,$ namely gradient, curl, and divergence. It is possible to treat the gradient of a differentiable function $f$ on $\mathbb{R}^3$ at a point $x_0$ as the Fréchet derivative at $x_0$. But it does not work for curl and divergence at all. Fortunately there is another abstraction that works fo...

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