How is differential form different from ordinary calculus objects?

Question

I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics objects. Can anyone explain this?

Answer 1

Differential forms is a way of formulating a calculus on manifolds without a strict adherence to coordinates. Besides differential forms being necessary to learn some differential geometry, its absolutely necessary for understanding Stokes' Theorem (modern version), which generalizes the classical Kelvin Stokes into manifolds and has the classical Kelvin Stokes, Green's Theorem and the Divergence theorem as easy corollaries.

Besides the technical help differential form provides, it also makes things spectacularly beautiful. For example, here is the modern Stokes' Theorem:

$$\int_{\partial \Omega} \omega = \int_{\Omega} d\omega$$

where $d\omega$ is the exterior derivative of the differential form $\omega$ and $\Omega$ is an orientable manifold.

Answer 2

Differential forms are a way to talk about fields that aren't vector or scalar fields--or at least, to talk about those and things beyond them. In traditional vector calculus, you only talk about scalar fields and vector fields. But using differential forms, you can talk about something like $\omega = 2y \, dx \wedge dy$ as a "bivector" field, a field of oriented planes throughout space.

You don't need to do this in 3d because each plane has a unique normal vector, so you can cheat and just use the vectors. But outside of 3d, you must do something like this to talk about all kinds of fields meaningfully. You can also talk about oriented volume fields and so on. Again, this is something looked over in vector calculus because there is only one unit volume in 3d, so you can pretend it's really a scalar field when it's not.