# Is there a good treatment of “familiar” physics using exterior calculus, AKA differential forms?

By familiar physics, I mean the physics of things I can reach out and touch. In other words, neither relativity nor analytical dynamics, etc.

After re-reading chapter 4 of MTW's Gravitation yet again, I am left wondering where the pay-off is. Similarly, reading through the introductory parts of Differential Forms with Applications to the Physical Sciences, by Harley Flanders, I have the impression that, to some extent, exterior calculus is a solution in search of a problem.

Among the purported advantages Flanders touts for exterior over tensor calculus is that tensor calculus loses the essential concepts in a "maze of indices". But, exterior calculus comes with its own, similar baggage. Ideas such as "pull-backs" render the familiar (reparameterization) intractable. Indeed, some of these complications in exterior calculus can be expressed using a comparable maze of indices. See, for example, Tensors, Differential Forms, and Variational Principles By: David Lovelock, Hanno Rund. Following the approach used in Advanced Calculus of Several Variables By: C. H. Edwards, Jr., I came up with some pretty ways of combining multi-indices and the Einstein summation convention which compellingly condense the maze of indices of exterior calculus.

But, thus far, all treatments and applications of exterior calculus I have encountered are fairly advanced. When I learned to use vectors, I didn't need an entire arsenal of theory in order to make productive use of them. I could have, and probably should have learned tensors in the same way. I can honestly say that I never really understood tensors until I applied them to such familiar topics as fluid flows and elasticity. In particular, the gradient of a velocity field opened my eyes.

Is there any treatment of "familiar" physics using differential forms which illustrates their applicability without adding the intellectual challenge of more esoteric topics such as the electromagnetic 2-form, or the sympletic geometry of phase space? It may be the case that such applications have not been made simply because it amounts to "reinventing the wheel". These topic have been effectively treated using the techniques of introductory physics, and are therefore uninteresting to the masters of differential forms. I am interested in such redundancy.

Any suggestions?

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• All of vector calculus (and hence all of E&M, fluid mechanics, etc.) can be formulated using differential forms, and there are plenty of pedagogical descriptions of it from a mathematical point of view. However, unless you are dealing with a spacetime that is more complicated than (3+1)-dimensional Minkowski, it is a quite a bit of additional effort for very little real improvement. – Buzz Dec 24 '18 at 23:39
• I'm not really interested in "improvement" per se. I am interested in gaining familiarity with differential forms in a realm in which my "naive" reasoning is applicable. That the Hodge dual of the 4-curent density equals the charge 3-form in Minkowski space isn't immediately accessible to my intuition. – Steven Thomas Hatton Dec 24 '18 at 23:54
• I disagree with @Buzz and would like to argue that the more common notation is unnecessarily clumsy and not to the point. There is a rather old book by Westenholz and the book by Nakahara on "Geometry, topology and physics" treats more modern applications. (But I should warn you: if you follow these, you probably won't be willing to look at any of the traditional expositions of, say, Maxwell's theory. ;-) – user178876 Dec 25 '18 at 1:13
• From what I have read differential forms are a special case of a tensor algebra with certain terms quotiented out to be zero. – Emil Dec 25 '18 at 8:32

## 2 Answers

$$\let\lam=\lambda \let\om=\omega \let\w=\wedge$$ Do you include thermodynamics within "familiar physics"? If so, I can't offer you a reference to printed books (although I can't exclude there is some). I developed such a treatment, almost 30 years ago, when teaching General Physics I, a full year, 1st-year (19) Physics course in the Pisa University (Italy). Lecture notes are available, but in italian.

I'll give you a sketch of a special argument, the proof of the following theorem:

If a thermodynamic fluid obeys the equation of state $$PV = RT \tag1$$ then internal energy depends on temperature alone.

(The proof is sketchy also in that some necessary previous steps are left understood.)

1) Two differential forms are known: $$\om$$ for heat, $$\lam$$ for work. The first principle is stated as $$\lam + \om = dU.$$

2) Work is $$\lam = -P\,dV$$, heat is $$\om = T\,dS$$. Then $$dU = T\,dS - P\,dV.\tag2$$

3) Differentiating (2) $$dT \w dS = dP \w dV.\tag3$$

4) Differentiating (1) $$P\,dV + V\,dP = R\,dT.$$ External product with $$dV$$ gives $$V\,dP \w dV = R\,dT \w dV.\tag4$$

5) From (1) we have $$V = RT/P$$ and substituting into (4) $$T\,dP \w dV = P\,dT \w dV$$ and using (3) $$T\,dT \w dS = P\,dT \w dV$$ $$dT \w (T\,dS - P\,dV) = 0$$ or, by (2) $$dT \w dU = 0.\tag5$$

6) It had previously been shown that if for two scalar functions $$f$$, $$g$$ one has $$df \w dg = 0$$ then $$f$$ and $$g$$ are functionally dependent, i.e. (loosely speaking) one is function of the other. Then (5) proves the theorem.

• I will have to thing about your example. On a similar note: I think a differential forms approach might be applicable to the difficult task of producing the "obvious" basics of kinetic theory of gases. hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/kinthe.html#c2 It may not reduce the amount of hand-waving, but it might provide an alternative view. Rather than talking about particles bouncing back and forth between walls, count the number of "walls" penetrated by the average molecule. Represent the "wall" density using a 1-form. – Steven Thomas Hatton Dec 26 '18 at 16:06

I wanted to suggest Flanders, but apparently you know it already and you did not like it. Anyway, looking at your questions, it seems that you are struggling with differential geometry in general.

For this reason I would like to suggest Fecko: Differential Geometry and Lie Groups for Physicists. It is quite good and a pleasant reading.