< 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).

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.