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I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical significance of it. Most books briefly introduce the exterior algebra and then go on to prove theorems about differential forms without ever geometrically motivating anything, and this does not help me to truly understand them. Could someone explain or give a reference to a text explaining differential forms more geometrically. I would very much like to have a deep understanding of them.

Thanks very much

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Here is a list of some references: – angryavian Jul 10 '13 at 22:03
up vote 8 down vote accepted

There are grave issues with the common thinking inherent to differential forms. For instance, in vector calculus, you can consider a vector field

$$\mathbf G(\mathbf r) = \frac{\mathbf r}{4\pi r^3}$$

You can meaningfully integrate the divergence of this vector field,

$$\int_{M} \nabla \cdot \mathbf G \, dV$$

and if your region contains the origin, you get 1, else 0.

In differential forms language, you have as your only hammer the exterior derivative. Instead of considering vector fields like you're used to from vector calculus, you have to convert $\mathbf G$ to a 2-form $\omega$.

$$\omega = \frac{z \, dx \wedge dy + y \, dz \wedge dx + x \, dy \wedge dz}{4\pi r^3}$$

And then you do an integral like

$$\int_{M} d\omega$$

as is usually done. But this is the big problem with learning differential forms when coming from vector calculus. You have to take all the stuff you're familiar with and turn it upside down. You have to take vector fields and turn them into two-forms to take meaningful divergences; you have to convert them to one-forms to take meaningful curls (though this is much less peculiar).

What's actually going on here can be explained in a different formalism called geometric calculus. It enforces that when we integrate over geometric regions (surfaces, volumes, and so on), the integration measure itself not only contains magnitude formation (the area of the surface, the size of the volume) but direction information, too. So instead of $dV$, we integrate over $d\mathbf V$, a 3-vector measure. Since all 3-vectors in 3d are scalar multiples of each other, we call this simply $\mathbf I \, dV$.

We can meaningfully integrate vector field divergences this way without arbitrarily converting them to two-forms. Rather, the "need" to do so follows from algebra.

$$\int_M (\nabla \cdot \mathbf G) (\mathbf I \, dV)$$

The 3-vector $\mathbf I$ turns directions to orthogonal planes and points to volumes. I submit to you, for reasons too complex to get into, that $(\nabla \cdot \mathbf G) \mathbf I = \nabla \wedge [\mathbf G \mathbf I]$ using the underlying rules of Clifford algebra that geometric calculus is built on top of. It is $\mathbf G \mathbf I$ that is the 2-form $\omega$, and it is $\nabla \wedge$ that truly represents, for all objects, the exterior derivative.

What this says geometrically is that there is an equivalence between the following: taking the divergence of a vector field (which yields a scalar) and converting that vector field to its orthogonal planes, finding the derivative orthogonal to all of those planes, and then converting the resulting volume back to a scalar. That is the nature of the divergence as seen in differential forms, and it becomes very clear using geometric calculus.

Geometric calculus is a natural and powerful extension to the vector calculus you probably already know. The notation is a lot more similar, surely, while still being able to fully express everything you would need to do in higher dimensions. It is called geometric calculus for a reason, as many of the main proponents try to emphasize, as much as possible, the geometric nature of the underlying algebra and the calculus.

If you become interested in this field, the canonical reference is Clifford Algebra to Geometric Calculus by Hestenes and Sobczyk. It's a slightly older text, but it contains good material on the relationship between multivector functions and differential forms. A more modern treatment is Vector and Geometric Calculus by Macdonald, which aims to be a more elementary text. Geometric Algebra for Physicists by Doran and Lasenby contains many applications and a good chapter on geometric calculus that condenses Hestenes and Sobczyk into something a little more approachable.

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Clifford algebra is beautiful and powerful, but a lot to ask of someone at the freshman/sophomore level learning multivariable calculus. Of course, a good first course should emphasize all the correspondences between vector fields and differential forms in $\mathbb R^3$. – Ted Shifrin Jul 10 '13 at 22:38
For such an audience I would recommend Macdonald's book (well, both his books, the first being Linear and Geometric Algebra, which introduces more of the algebraic aspects). They are lean and efficient at what they do and in maintaining solid contact with traditional matrix algebra, vector algebra, vector calculus, and other more traditional approaches. – Muphrid Jul 10 '13 at 22:40

Here are a couple of references that you might find helpful.

Vector Calculus, Linear Algebra, and Differential Forms by Hubbard and Hubbard. This book is written for undergraduates and integrates (pardon the pun) the study of multivariable calculus, linear algebra, and finishes with differential forms. If you'd like intuition for the geometry of forms in $\mathbb{R}^3$, without too much high-level symbol-pushing, this is a good place to look.

The Geometry of Differential Forms, by Morita, is a monograph which starts with basic definitions and proceeds to describe the utility of differential forms in various contexts, including (if my memory serves) Hodge theory and bundle-valued forms.

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I'll also toot my own horn and add my own book Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds, which I wrote when my students couldn't penetrate Hubbard and Hubbard :) Differential forms come about by adding layers of calculus to the algebra of determinants, which compute oriented $k$-dimensional volumes.

I would also suggest Harley Flanders's Differential Forms with Applications to the Physical Sciences (now in Dover). At a higher level, especially if you're interested in differential geometry and physics, look at a surprisingly concrete exposition, R.W.R. Darling's Differential Forms and Connections.

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