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0

I think you want $$(\varphi_1 \wedge\dots \wedge\varphi_k)(v_1,\dots,v_k)=\frac{1}{k!}\sum_{\sigma\in S_k}sgn(\sigma)\varphi_1(v_{\sigma(1)})\varphi_2(v_{\sigma(2)})\cdots\varphi_k(v_{\sigma(k)}).$$

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Good question! Here's a start. The ordinary derivative in one-variable calculus is a Lie derivative along a special vector field on $\mathbb{R}$; in particular, it is not a special case of the exterior derivative. The exterior derivative is instead some kind of "universal derivative": it records all of the information you would need to determine the ...

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That $d^2 = 0$ is probably something you were already taught in vector calculus. For instance, you probably remember that $\nabla \times \nabla \phi = 0$ for $\phi$ a scalar field, or that $\nabla \cdot (\nabla \times E) = 0$ for $E$ a vector field. It's a good exercise to show that both of these can be written as $d^2 f = 0$ and $d^2 E = 0$. Of course, ...

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