Can you give me some good books on differential forms?

>>8591030
The course on them at my school used "Vector Analysis" by Klaus Jänich.

Know linear algebra and basic topology before going into it.

>>8591030
do Carmo - Differential Forms and Applications
Flanders - Differential Forms with Applications to the Physical Sciences
Lovelock - Tensors, Differential Forms, and Variational Principles
Harold M. Edwards - Advanced Calculus: A Differential Forms Approach

I liked Spivak's Calculus on Manifolds. Also Singer and Thorpe's Elementary Topology and Geometry gives a decent intro and has a bunch of other cool info you can go through later on.(De Rham Cohomology, Curvature, Gauss Bonnet Theorem, etc.)

>>8591376
Spivak only works with manifolds with manifolds viewed as subsets of R^n which is kind of annoying.

>>8591046
>>8591196
>>8591376
Followup question since no. 1 mentioned topology:

what topics should I know before dealing with differential forms?

My skills:
>Linear Algebra
>Vector Calculus
>Ordinary and Partial Differential Equations

>>8591520
Basic topology.

i.e. Definition of a top. space, homeomorphisms, connected spaces, compactness, Hausdorff spaces, etc.

That way you can learn the proper definition of a manifold.

>>8591030

Tensor Analysis on Manifolds (Dover Books on Mathematics) by Richard L. Bishop
Geometrical Methods of Mathematical Physics by Bernard F. Schutz

>>8591520

You can either do it right and be done with it OR half ass it and constantly have to wrestle with shoddy knowledge.

If you want a good foundation for future math studies, then read:

Book of Proof by Hammack (Free: http://www.people.vcu.edu/~rhammack/BookOfProof/)
Linear Algebra (Dover Books) by Shilov
Elementary Real and Complex Analysis (Dover Books) by Shilov
Elementary Topology (Dover Books) by Gemignani

>>8591520
Why are you dealing with differential forms? Are you studying differential manifolds / physics?

>>8592402
I'm doing a bachelors in mech eng.
but I am going to do a Ph.D. in applied Math and I wanted to catch up

>>8592532
I guess you can skim through 'Quick introduction to tensor analysis' and 'Course of differential geometry' by Sharipov(free online) to understand that tensors are things represented by arrays that 'behave' under change of coordinates, and then realize that the coordinates 'behave' if and only if the corresponding multilinear form is well defined(hence the usual representation). Then you can move to the first ~30p of Do Carmo's book, which explain differential forms on R^n, where the tangent space to a point is just a copy of R^n. Here to prove Poincare's lemma for 1-forms (homotopy version) you will need to know what does it mean for two curves to be homotopic, and you have to use Lebesgue's lemma hence you need to know what compactness is(so yeah, a bit of standard topology is usefull if you don't want to restrict yourself to star-shaped domains or ugly things like that). At this point you can rewrite the gradient, curl and divergence in term of differential of forms, and using Poincare's lemma deduce div rot = 0 and rot grad = 0. To move on differential forms on differential manifolds you will now need to understand what the tangent space is, link the definition in terms of class of curves to the one of derivations, what is the differential(aka pushforward) of a differentiable map and then you can transfer the differential calculus of R^n to the manifold by means of charts.
Do Carmo is really hasty(even for the first part), so I would suggest reading 'Introduction to smooth manifolds' by Lee, and then to go deep(where I haven't been) 'Geometry of differential forms' by Morita. Another useful resource I once found was Patrick Hafkenscheid's bachelor thesis 'De Rham Cohomology of smooth manifold'.
Notice that differential forms only need a vector space to be defined, so you can develop them in an 'axiomatic' way, and use for example in algebraic geometry.