This was how differential forms were introduced in the course. I understood all of this from an algebraic standpoint, but I was lacking any geometric intuition for differential forms whatsoever. Say you have a k-form on some manifold and you evaluate it at some point which gives you an alternating covariant k-tensor. Then when you evaluate that at k tangent vectors at the point you get a scalar, does this scalar have any geometric meaning? Does it measure anything? Later when we did Riemannian manifolds and introduced the volume form that was at least a little more intuitive, as far as I remember, but general differential forms were intuitively a complete mystery to me. Also I kind of got their usefulness in an algebraic sense when we did some typical vector calculus calculations using the concepts of divergence and curl, but I didn't have much intuition for these concepts since I don't have a physics background and only worked with vector fields in this abstract setting. Unfortunately we did not cover De Rham cohomology. Thanks for your answer, I will take a look at planimeters.
https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2for...
The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.
Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".
There's some other uses. They form the basis of De Rham Cohomology [2] which is a useful and computational way of describing topological properties of a manifold (recall how Stokes's theorem and friends show how the topology of a space constrains the integrals of differential forms).
Another thing is that they're specific kinds of tensors on the manifold. Tensors represent basically all the information we might be interested in about a manifold (for example, its curvature). Differential 1-forms are "dual" to vectors and are therefore important to building more complex tensors (higher tensors take in some number of vectors and 1-forms and output a value).
And just as regular integration and differentiation relates to solving of differential equations, differential forms are needed for differential equations that are on a manifold.
[1] https://en.wikipedia.org/wiki/Planimeter [2] https://en.wikipedia.org/wiki/De_Rham_cohomology