is created by David Witten, a mathematics and computer science student at Vanderbilt University. For more information, see the "About" page.

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 $VF^{\infty}(U) \to \Omega^{n-1}(U)$


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