From what I understand one of the main benefits of differential forms over Riemann integrals is that you're supposed to be able to integrate differential forms without parametrizing your curve (or surface/ etc) but all of the examples I've seen of people using differential forms so far have included parametrizing the curve (or surface/ etc). Assuming my understanding of differential forms is not wrong, can someone provide an example of how to use them to calculate an integral without first parametrizing the region of integration?
2026-04-06 03:45:31.1775447131
Evaluating an integral without a parametrization
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I think maybe you've misinterpreted what someone has said about integration. What's true is that the integral of a differential $k$-form over a $k$-manifold has a well-defined meaning, independent of any particular choice of parametrization. In other words, if you choose one parametrization and I choose another, we'll get the same answer for the integral.
However, in general, I don't know of any way to compute the integral of a differential form, other than by choosing a parametrization and working it out.
In some special cases, there are shortcuts that might allow you to avoid working directly with a parametrization. For example, Stokes's theorem often gives you a way to show that an integral is zero, or that the integral of a form over one manifold is equal to its integral over another simpler one, and if you're lucky this can obviate the need to do any computations with parametrizations. But those shortcuts work only in very special cases.