11–20 of 48 posts
I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth m…
It builds up the theory of differential geometry using Scheme, just like SICP and SICM, and is a fantastic read for programmers wanting to grasp the topic in a more familiar language.
It is available for download for free on the publisher's website: https://mitpress.mit.edu/books/functional-differential-geome...
This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it needs to be able to compute the positive and negative components and take the difference, which being finite to make sense, must be absolutely finite) and is distinct from the Henstock-Kurzweil integral which works directly with a tagged partition of the underlying set (which implies an orientation) and is able to integrate (some) functions which are not absolutely integrable.
Nevertheless, moving to differential manifolds introduces problems for the Henstock-Kurzweil integral becames the local orientation in charts does not always induce a sensible global orientation. However, integration on a manifold isn't so much integration of a function of several variables as it is integration of several functions of several variables. That one needs additional machinery to deal with the "several functions" part is unsurprising.
I can't recommend studying the Henstock-Kurzweil integral enough! Strangely enough, despite being more general, it is far more approachable than the Lebesgue integral.
https://www.amazon.com/Princeton-Companion-Mathematics-Timot...
a great (even if expensive) math book
I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth m…
I think some grassmannian computations would be good in this context but, on the other hand, they would become very cumbersome very soon.
As someone says below: Spivak's Differential Calculus on Manifolds is exceptionally good.
I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong, but an orientation seems to always be implied in being able to compute a Riemann sum over a tagged partition.) This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it need…
I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong, but an orientation seems to always be implied in being able to compute a Riemann sum over a tagged partition.) This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it need…
this is his article in the princeton companion to mathematics https://www.amazon.com/Princeton-Companion-Mathematics-Timot... a great (even if expensive) math book
I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth m…
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