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.