Resources showing how to use differential forms in Physics

Question

I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently from vector calculus, I'm not being able to grasp how to use those objects in Physics.

For instance, I know that a $k$-blade represents an oriented piece of $k$-dimensional vector subspace from a vector space, I know that $k$-forms represents ways of "measurements" of those multivectors, and all of that stuff, but I simply don't know how to use them in practice like I've learned in the past with dot product, cross product, integration on the plane and space and so on.

Is there any book ou there that show those things in a suitable way for a physicist? I know that there are many books good for mathematicians out there, but they mainly focus on proving consequences of the definitions, rather than applying them. I'm searching for something that shows how to really apply in Physics all that elements from calculus on manifolds.

What I really mean is the following: most mathematics books on the topics tell how to prove theorems only. And many times I see the exposition and think "it's not possible to use this in practice" while there are tons of interpretations and usages in Physics. I just didn't find yet a book that show this.

Just an example of what I say: in electrostatics we start with Coulomb's law that gives us the electric force between two charges. From that and the superposition principle we get an expression for the electric field. All of this is done with vectors, so that when we study those objects we use divergence, curl, and all those machinery from vector calculus. It seems straightforward to represent those things as vectors, but it doesn't seem obvious to do so with forms. Indeed, I don't even know how to write this in terms of forms without writing first with vectors and then transforming with a metric.

EDIT: I'm mainly searching for resources that covers this topic in the lines that vector calculus is seem on Arfken's "Mathematical methods for physicists" book and the introductory chapters on Griffith's "Introduction to Electrodynamics" and Marion's "Classical Dynamics of Particles and Systems". I don't know if this kind of resource can help, but I thought this edit could make my question more specific.

Answer 1

Kyle Kanos mentioned Geometric Algebra for Physicists. While geometric algebra is somewhat different in notation from differential forms, the basic concepts are all there, and in many ways, geometric algebra avoids some cumbersome things that differential forms does (I'm thinking of Hodge duality in particular). I think the notation is easier to relate to vector calculus than with differential forms.

The book itself is an intermediate level introduction to geometric algebra, with significant emphasis on applications. The first few chapters deal with the basics: the algebra itself, multiplication operations, linear algebra, and calculus of multivector fields.

All of these are essential to the applications that are discussed. Electrodynamics and special relativity form the first major topic. Having shown the form of rotations using quaternion-like rotors in 3d, the special relativity passages make good use of a similar construction for Lorentz boosts. The EM field as a bivector is also discussed, as well as the form of Maxwell's equations using geometric calculus and the spacetime algebra.

The book then goes on to explore applications toward quantum mechanics. I think this was a major focus of the authors, but to me, this section was considerably less clear or elegant. The authors confine themselves to treating spin-1/2 particles, showing that the algebra of spin-1/2 is identical to the spacetime algebra, and that spin operations can be understood geometrically. All well and good, but I think this is quite insufficient, and some consideration or treatment of how GA could be used broadly for quantum mechanics would've been warranted.

On the tail end of this is a sojourn into general relativity, motivated in part by trying to generalize the Dirac equation. Thus, rather than classic GR proper, we get something like an alternative to Cartan formalism. I think this is a point of significant interest, as the use of GA here manages to demystify a lot of tensor manipulation inherent to GR. This section contains a well-worked, extended example to spherically symmetric spacetimes, as well as a brief discussion of axially-symmetric spacetimes, so both Schwarzchild and Kerr black holes are considered, along with stars and cosmic strings.

The book is somewhat sophisticated. The introductory chapters could be reasonable for learning GA (and some of the concepts that apply to differential forms; some of the more mathematically inclined passages on geometric calculus detail at length the relationship between multivector fields and differential forms), but I would consider lighter introductory material for a first reading. Alan MacDonald's books on GA and GC are more mathematically focused, but they're also targeted at undergrads, so for pure mathematical foundation, I think they're a better start. For applications, GA for Physicists is very sound, touching electrodynamics, quantum, and general relativity, with a smattering of rigid body dynamics (which is vastly simplified using rotors from GA).

Answer 2

If you'd like to quickly obtain an understanding of the basics of differential forms, including their relation to connections, tangent bundles etc. I recommend the first 4 online lectures of the Perimeter Institute from the Gravitational Physics course (13/14, R. Gregory).

Answer 3

For a start, you might have a look at the paper "Differential forms for scientists" by J.B. Perot, Journal of Computational Physics, 2014, Vol. 257 Part B, I found it useful.