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I am recently learning some basic differential geometry. As I understand, differential forms provide a neat way to deal with the topics in calculus such as Stoke's theorem. In order to define the differential forms, one needs to learn the basic concepts in multi-linear algebra.

Here are my questions:

  • Are "differential forms" basically an algebraic approach to (multivariable) calculus?
  • If the answer is YES, what would be a analysis counterpart and what are the advantages and disadvantages of these two approaches?
  • If the answer is NO, how should I understand it properly? (EDITED: Are differential forms the only approach to multivariable analysis calculus?)

MOTIVATION:

When I look for a rigorous theoretical approach to multivariable calculus, such as I asked in this question , differential forms are almost always included in the books recommended. I am thus wondering if this is the only approach to multivariable calculus.

I may not being asking a good question. Any suggestion for improving the questions above will be really appreciated.

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2  
I'm not sure what you mean. What would it mean to say yes to this question? What would it mean to say no? –  Qiaochu Yuan Aug 28 '11 at 0:46
    
@Qiaochu: Oops, I should have made it more precise question. I am editing it. –  Jack Aug 28 '11 at 0:49
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Okay. I am not sure in what sense differential forms don't count as analysis. I would say that they are an elegant blend of algebra and analysis. Perhaps the answers to mathoverflow.net/questions/10574/… might be helpful to you. –  Qiaochu Yuan Aug 28 '11 at 1:24

2 Answers 2

up vote 8 down vote accepted

Any approach to multivariable calculus necessitates some modicum of multilinear algebra ( for instance because the change of variables formula for integrals uses determinants), and the cleaner the interface between calculus and algebra, the better. The language of differential forms is a clean interface between the two, and this language generalizes from $\mathbb{R}^n$ to arbitrary manifolds. That is, differential forms are how calculus is done on manifolds. Or, to be more accurate, they are a convenient algebraic formalism for doing calculus on manifolds. To abandon differential forms is not to go to analysis, but to lose oneself in an ocean of notation.

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4  
Your "for instance" is the reason for multilinear algebra in multivariable calculus. –  KCd Aug 28 '11 at 3:11
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@AmbroseH: I don't understand your last two sentences... You seem to imply that there is another way of doing calculus on manifolds, one that involves an "ocean of notation." As for myself, I don't know of any other approach to integration on manifolds (except maybe if one counts densities as something separate), do you? –  Jesse Madnick Aug 28 '11 at 4:25
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The "ocean of notation" is probably related to Ricci-calculus and tensor analysis. But both Ricci-calculus and tensor analysis seem to be quite successful, and there are definitively situations that can be described by tensors but not by (alternating) differential forms alone. But there are also things like "Christoffel symbols" that don't even fit into the formalism of tensor analysis (or Ricci-calculus), and evoke an even worse "ocean of notation". (Conclusion: differential forms are a good first step in avoiding an "ocean of notation", but differential geometry has many more notions.) –  Thomas Klimpel Aug 28 '11 at 14:54
    
@KCd: Duly noted. –  Robert Haraway Aug 29 '11 at 16:16
    
@KCd I think there is more motivation for multilinear algebra in multivaraible calculus than just the change of variables formula for integrals. For instance, the $kth$ derivative of a smooth map from $\mathbb{R^n} \to \mathbb{R}$ is really a symmetric $k-$tensor field on $\mathbb{R}^n$. Undergrads even see a little bit of this, with the Hessian "matrix" (really $2$ -tensor). –  Steven Gubkin Feb 12 at 19:00

The answer to the question

  • Are differential forms the only approach to multivariable calculus?

is a definitive NO. Differential forms are a topic of differential geometry or calculus on manifolds. And I think Wikipedia takes the right approach to calculus on manifolds by first talking about implicit and inverse function theorems, vector fields, the directional derivative, the Lie derivative, the Lie bracket and other important topics before even mentioning differential forms.

If you define multivariable calculus as the extension of calculus in one variable to calculus in more than one variable, and divide calculus into differential calculus and integral calculus, an introduction to differential forms as part of multivariable integral calculus could make sense.

  • Are "differential forms" basically an algebraic approach to (multivariable) calculus?

It's true that differential forms have important algebraic properties that are useful for the global analysis of manifolds:

  • Differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. (A vector field can only be pulled back by an immersion if it's tangential to the immersed manifold.)
  • The exterior derivative has the important property that $d^2 = 0$.

This leads to exact sequences which allow to build algebraic cohomology theories.

  • What would be a analysis counterpart and what are the advantages and disadvantages of these two approaches?

One drawback of differential forms for high dimensional manifolds comes from the curse of dimensionality. The vector space of alternating $k$-linear forms over an $n$-dimensional vector space has dimension $\frac{n!}{k!(n-k)!}$. So failing to visualize differential forms "correctly" before your inner eye is not the fault of your imagination, but simply one of the disadvantages of differential forms. As a consequence, there is nothing wrong with visualizing differential forms in $\mathbb{R}^3$ as vector fields.

Perhaps surprisingly, there are ways around the curse of dimensionality. (Monte Carlo methods are probably the best known examples, but there also exist deterministic methods.) However, there is so much material to be covered in a multivariable calculus course that the curse of dimensionality is rarely even mentioned at all.

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