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I like the Geometric Algebra approach to incorporating differential forms into physics that is taken by Dr. David Hestenes and contained in his numerous works over the last few decades but see no mention of Geometric Calculus here. Are you familiar with it?

"DIFFERENTIAL FORMS IN GEOMETRIC CALCULUS by Dr. David Hestenes

Abstract: Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant application of the system is to relativistic physics where it is referred to as Spacetime Calculus. The fundamental integral theorems are discussed along with applications to physics, especially electrodynamics."

I first encountered differential forms in the classic GRAVITATION by Misner, Thorne and Wheeler but I later found that electrical engineers I knew considered it to be a fancy theoretician's formalism impractical for everyday use.Yet, differential forms have become more and more popular it seems and some of the more modern introductory texts on vector analysis have a chapter on differential forms.

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"DIFFERENTIAL FORMS IN GEOMETRIC CALCULUS by Dr. David Hestenes

Abstract: Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant application of the system is to relativistic physics where it is referred to as Spacetime Calculus. The fundamental integral theorems are discussed along with applications to physics, especially electrodynamics."

I first encountered differential forms in the classic GRAVITATION by Misner, Thorne and Wheeler but I later found that electrical engineers I knew considered it to be a fancy theoretician's formalism impractical for everyday use.Yet, differential forms have become more and more popular it seems and some of the more modern introductory texts on vector analysis have a chapter on differential forms.

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