So, the standard interpretation of a differential 1-form's integral over a compact, connected, oriented 1-manifold is work, and the standard interpretation of a differential 2-form's integral over a compact, connected, oriented 2-manifold is flux, in engineering parlance.
What would be an interpretation of a differential 3-form's integral over a compact, connected, oriented 3-manifold (say, a submanifold of the 6D c-space of a robot arm) in standard engineering parlance?
I'm trying to motivate differential forms as the generalizations of tangent vector fields over curves and normal vector fields over surfaces in Calc III to my Calc III students, but it's difficult to motivate something without knowing its interpretation. (I only actually assign problems on 3- and higher-form integrals as Extra Credit problems, but present the notions of 0-, 1-, 2-, and 3-forms on open, connected subsets of $\mathbb{R}^2$ and $\mathbb{R}^3$, their wedge products, their exterior derivatives, and their Hodge stars as a parallel and unifying theory to the notions of functions, vector fields, cross products, and grad, curl, and div as part of the standard lecture. I also illustrate 0- and 1-forms and their Hodge stars on $\mathbb{R}$ to explain why there "is no" de Rham cohomology of open, connected subsets of $\mathbb{R}$ and why continuous 1-forms [Hodge stars of functions] therefore always have an antiderivative on compact intervals in $\mathbb{R}$ as part of FTC ["Exterior derivatives are always zero and points separate"].) Any actual examples of integrals of 3- (and higher-)forms in engineering work to illustrate would be GREATLY appreciated.
[Here is the Extra Credit problem http://airvigilante194.sdf.org/Scripts/ec.pdf.
Here are some scripts to solve the last two problems https://library.wolfram.com/infocenter/MathSource/482/ http://airvigilante194.sdf.org/Scripts/ec-p23Form.nb http://airvigilante194.sdf.org/Scripts/ec-p2DiffForm.wxmx http://airvigilante194.sdf.org/Scripts/ec-p3DiffForm.wxmx.]