Definition 6.5.1 (Work form). The work form $W_{\vec F}$ of a vector field $\vec F = \begin{bmatrix} F_1\\ \vdots \\ F_n \end{bmatrix}$ is the 1-form defined by

$$ W_{\vec F}(P_x(\vec v)) = \vec F(x) \cdot \vec v. $$

In coordinates, this gives $W_{\vec F} = F_1 dx_1 + \dots + F_n dx_n$. For example, the one-form $ydx - xdy$ is the work form of $\vec F = \begin{bmatrix} y \\ -x \end{bmatrix}$.

This is equivalent to how we understand work done by forces in physics.

Definition 6.5.2 (Flux form). The flux form $\Phi_{\vec F}$ is the 2-form field $\Phi_{\vec F}(P_x(\vec v, \vec w)) = \det [ \vec F(x), \vec v, \vec w]$.

In coordinates, this is $\Phi_{\vec F} = F_1 dy \wedge dz - F_2 dx\wedge dz + F_3 dx\wedge dy$. in $\R^3$, this shows us the flow of a vector field through a parallelogram: $\vec F(x) \cdot (\vec v \times \vec w)$.

Definition 6.5.4 (Mass form of a function). Let $U$ be a subset of $\R^3$ and $f: U \to \R$ a function. The mass form $M_f$ is the 3-form defined by

$$ M_f(P_x(\vec v_1, \vec v_2, \vec v_3)) = f(x) \det [\vec v_1, \vec v_2, \vec v_3]. $$

It is natural to think of this as integrating a density function over a manifold.

Definition 6.5.7 (Flux). The flux of a vector field $\vec F$ over an oriented surface $S$ is $\int_S \Phi_{\vec F}$.

Definition 6.5.10 (Flux form on $\R^n$). If $\vec F$ is a vector field on $U \subset \R^n$ and $\vec v_1, \dots, \vec v_{n-1}$ are vectors in $\R^n$, then the flux form $\Phi_{\vec F}$ is the $(n-1)$-form defined by

$$ \Phi_{\vec F} P_x(\vec v_1, \dots, \vec v_{n-1}) = \det [\vec F(x), \vec v_1, \dots, \vec v_{n-1}]. $$

Definition 6.5.11 (Mass form on $\R^n$). Let $U$ be a subset of $\R^n$. The mass form $M_f$ of a function $f: U \to \R$ is given by

$$ M_f = fdx_1 \wedge \dots \wedge dx_n. $$