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Dictionary Between Vector Calculus and Differential Forms

General Idea

This dictionary will convert ideas in vector calculus ("vector field”, “flux”, “etc.”) into differential forms. This also gives us a geometric intuition of what’s going on.

A bit of terminology: $C^{\infty}(U)$ is the set of all smooth functions from $f: U \to \mathbb{R}$. Let $VF^{\infty}(U)$ be the set of all smooth vector fields F: $U \to \mathbb{R}^n$. $\Omega^k(U)$ denotes the set of all differential k-forms.

Starting Point

Our starting point is at $\Omega^{0}(U)$, which is a normal equation. Therefore, it's equal to $C^{\infty}(U)$

Work Form

$$W_F = W_{\nabla f} = df$$ By definition, $df = f_xdx + f_ydy + f_zdz$. $$\nabla f = \begin{bmatrix}f_x\\f_y\\f_z\end{bmatrix}$$ $$W_{\nabla f} = f_xdx + f_ydy + f_zdz = df$$ So, the work form goes from $VF^{\infty}(U) \to \Omega^1(U)$.

Flux Form

The flux form is written like this: $$\phi_{F(x)}(v_1, ... v_{n-1}) = \det{\begin{bmatrix}F & v_1 & v_2 & v_3 & ... & v_{n-1}\end{bmatrix}}$$ By the definition of differential form, this is part of $\Omega^{n-1}(U) $.

So, the flux form goes from $VF^{\infty}(U) \to \Omega^{n-1}(U)$

Curl

This is specific to three dimensions, but differentiation from the 1-form to the 2-form is equal to taking the curl of the gradient. $$f(x,y,z) = xy + yz^2 + xz^2$$ $$\nabla f = \begin{bmatrix}y + z^2 \\ z^2 + x \\ 2zx + 2zy \end{bmatrix} \to W_{\nabla f} = (y + z^2)dx + (z^2 + x)dy + (2zx + 2zy)dz$$ We know from before that $df = (y + z^2)dx + (z^2 + x)dy + (2zx + 2zy)dz$. Let's take the derivative again. $$d^2f = 1dy \wedge dx + 2z dz \wedge dx + 2z \wedge dy + dx \wedge dy + 2z dx \wedge dz + 2z dy \wedge dz $$ $$d^2f = (1 - 1)dx \wedge dy + (2z - 2z)dx \wedge dz + (2z - 2z) dy \wedge dz = 0$$ If we take the curl of $\begin{bmatrix}y + z^2 \\ z^2 + x \\ 2zx + 2zy \end{bmatrix}$, we also get 0.
David Witten

Manifolds