I Some Questions on Differential Forms and Their Meaningfulness

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I've been trying to get a meaningful understanding of the benefits of using differential forms. I've seen examples of physics formulas that are reduced to a very simple declarative form relative to their tensor counterparts. However to me it just seems like a notation change to implied tensor indices.

Some texts will say to do any actual computation you need to convert the differential form to tensor notation and work the problem through to completion.

Are there some deeper theorems or examples that show the superiority of differential forms over tensor methods?

Some of the books I've looked through are Flander's book, Wheeler's Gravitation and Hsu Vector Analysis Outline series book.

There's also a series of videos on Youtube most notably by Dave Metzler (others are classroom based tutorials):


In Metzler's tutorial he relates the various forms to their vector counterparts which shows some interesting connections between the forms themselves but then what.

I'm looking for some example that shows you gain an intuitive understanding of some system by looking at its differential form description.
 
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fresh_42

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To me it appears as a matter of elegance. E.g. ##\int_G \, d\omega = \int_{\partial G}\omega ## can hardly be expressed any better. What I do not understand is your implicit assumption of equivalence. The differential forms are tensors where the antisymmetry still has to be factored out, so they are not exactly equivalent, i.e. one is oriented the other is not. Furthermore are differential forms geometric in a way and tensors are more algebraic, the basic difference between an object and its coordinate form. But that's a mathematical point of view, let's see what physicists have to say.
 

Orodruin

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Are there some deeper theorems or examples that show the superiority of differential forms over tensor methods?
I am not sure what you are trying to say here. Differential forms are (a particular form of) tensors, namely completely asymmetric type ##(0,p)## tensors.
 
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Okay so my problem is why would I want to use differential forms over tensor methods?

A physics example might be using Lagrangian methods over using Newtonian force methods to solve a problem.
 

martinbn

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Okay so my problem is why would I want to use differential forms over tensor methods?
They already said it twice, that differential forms are tensors. So if you are using differential forms you are using tensors. That's why your question is unclear.
 
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Thanks for the answers. I guess I'm trying to understand my personal compelling reason for using Differential Forms and have yet to find it. I'll have to give it some more thought.
 

Orodruin

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Here are some compelling reasons: The language of differential forms ...
  • ... naturally describes the infinitesimal volume spanned by tangent vectors on a manifold.
  • ... generalise the concepts of gradient, divergence, and curl into a single concept, the exterior derivative.
  • ... it is sometimes the only way of obtaining a dual vector as a derivative of a scalar function that can properly map tangent vectors to scalars. The gradient is naturally a one-form.
  • ... is the natural way of defining integrals over manifolds and submanifolds.
  • ... as @fresh_42 already said $$\int_V d\omega = \oint_{\partial V} \omega$$ which is the second fundamental theorem of calculus, Green's theorem in the plane, the divergence theorem, the curl theorem, and many more all rolled up into one.
 
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Thanks, I've seen all those reasons too but I thought there was still something I wasn't understanding. The generalization with the exterior derivative looked to me as the most compelling reason but I thought maybe there was some computation aspect that favored Differential Forms over tensor methods.
 

WWGD

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Another benefit I have not seen mentioned: The use of DeRham Cohomology . Given a topological space, knowing the general characterization of forms -- specifically which are closed, which exact, actually gives you informatiuon about the _Topology_ of the space, since DeRham Cohomology group is isomorphic to other ("standard/Regular, meaning they satisfy a set of axioms) homological theories.
 

fresh_42

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I'm not sure why people are saying your question is confusing, it is not. The difference is just the notation of differential forms vs conventional tensor notation. I personally enjoy using the notation of differential forms and I'm not sure why we're still teaching electromagnetism without incorporating them... The place where I see that differential forms have a computational efficiency over conventional tensor notation is when dealing with Cartan Formalism. Actually, I enjoy using differential forms notation even when doing general relativity! But at the end of the day, it's just notation. You'll get the same results, as i'm sure you know.

This book is a good and easy introduction that incorporates physics: https://www.amazon.com/dp/1466510005/?tag=pfamazon01-20 (but this doesn't get into the Palatini formalism).

If you don't need to learn differential forms at this level this book+want physics, you will want to check out this appendix: https://cds.cern.ch/record/1517921/files/978-88-470-2691-9_BookBackMatter.pdf (It comes from the wonderful book of http://www.springer.com/la/book/9788847026919)
 

lavinia

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Since their was no response to @WWGD's post #9, I'd like to repeat it.
Another benefit I have not seen mentioned: The use of DeRham Cohomology . Given a topological space, knowing the general characterization of forms -- specifically which are closed, which exact, actually gives you informatiuon about the _Topology_ of the space, since DeRham Cohomology group is isomorphic to other ("standard/Regular, meaning they satisfy a set of axioms) homological theories.
.

DeRham cohomology theory expresses the real(or complex) cohomology of a smooth manifold in terms of the cochain complex of differential forms.

This theorem is not directly related to the way a differential form looks in local coordinates. It is more related to the integrals of closed forms over closed smooth submanifolds . Such integrals are unchanged under a change of gauge. That is: they are unchanged if the form is modified by an exact form. So De Rham theory does not care about the particular local representation as a tensor - but rather to the equivalence class of the form under a change by an exact form.

To prove De Rham's Theorem, one must think of differential forms as homomorphisms from the free abelian group of smooth singular chains into the real numbers. The value that this homomorphism takes on a smooth simplex is the integral of the form over the simplex. So this is a reconceptualization of forms as linear operators on a free abelian group. This idea does not seem to arise naturally from local tensor coefficients but rather from integration.

Given De Rham's theorem one can go in the reverse direction and ask whether important cohomology classes can be represented by interesting closed differential forms. A classic example is the Weil homomorphism which produces a collection of differential forms in the curvature 2 form matrix of a connection on a smooth principal Lie group bundle. These forms represent characteristic classes e.g. Chern classes which are important invariants of the bundle. Thus Chern classes which are defined topologically, are reinterpreted in terms of the differential geometry of the bundle. In other words, one has a link from the theory of characteristic classes of a bundle to its Differential Geometry. In this case, the particular representation of the form - and therefore its local representation as a tensor - is not unique since its cohomology class is independent of the connection. More generally the image of the coefficient map ##H^{k}(M;Z) →H^{k}(M;R)## is represented by differential forms whose periods are all integers which by itself is an important property of a closed form.
 
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atyy

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There's also a series of videos on Youtube most notably by Dave Metzler (others are classroom based tutorials) ...
Thanks! I had my first encounter with differential forms years ago in an inspiring class taught by Dave Metzler!

Okay so my problem is why would I want to use differential forms over tensor methods?
Differential forms allow one definition of volume and integration, even in the absence of a metric. Forms, and the wedge product for multiplying forms, are an abstraction of the concept of volume, which one can encounter in vector geometry as the area of a parallelepiped being the cross product of its sides.

Differential Geometry in Physics
Marián Fecko
http://davinci.fmph.uniba.sk/~fecko1/referaty/regensburg_2007.pdf
"Forms in L enter the play naturally when (oriented) volumes of parallelepipeds ... The tensor product α⊗β of two forms is not a form (just a tensor). If one, however, projects out the antisymmetric part of the result, one obtains a form."

Differential Forms and Integration
Terry Tao
https://www.math.ucla.edu/~tao/preprints/forms.pdf
"The concept of integration is of course fundamental in single-variable calculus. Actually, there are three concepts of integration ...

When one moves from single-variable calculus to several-variable calculus, though, these three concepts begin to diverge significantly from each other. The indefinite integral generalises to the notion of a solution to a differential equation ... The unsigned definite integral generalises to the Lebesgue integral, or more generally to integration on a measure space. Finally, the signed definite integral generalises to the integration of forms which will be our focus here. ...

The integration on forms concept ... yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds."
 
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stevendaryl

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Differential forms allow one definition of volume and integration, even in the absence of a metric.
My understanding of differential forms is pretty superficial. My confusion is this: When you change variables in an integral, say from ##x,y## to ##r, \theta##, you can't simply replace ##dx\ dy## by ##dr\ d\theta##. You have to include a factor of ##\sqrt{|g|}## where ##|g|## is the absolute value of the determinant of the metric in the new coordinates, which is ##r^2## for 2-D polar coordinates.

However, in terms of differential forms, you don't need the metric. So that's what I don't understand. Or is it just that if you want to integrate a scalar function ##f(x,y)##, you need the metric, but you don't need it if you're integrating a tensor field (of the appropriate type)?
 

Orodruin

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The point is that the regular volume form on a Euclidean space is the Levi-Civita symbol ##\epsilon_{ijk\ldots}## (##\ldots## for possibly higher dimensions) in Cartesian coordinates. However, the Levi-Civita symbol entries are not the components of a tensor with respect to general coordinates, they are the components of a tensor density. To find out what the volume form is in a general system, you therefore need to multiply by a scalar density of the appropriate weight that is equal to one in a Cartesian coordinate system. That something turns out to be the square root of the metric determinant (the metric determinant is a scalar density of weight 2 and you need a scalar density of weight 1). The argument can be made general to define a reasonable volume form with respect to the metric of any (pseudo-)Riemannian manifold.

Of course, as @atyy and @jedishrfu says, you can define volume integration without a metric, but when you do have a metric and want to interpret the volume form in terms of the volume of a small cube with some given sides, then there is only one choice for the volume form ##\sqrt{g} \epsilon_{ijk\ldots}##.

Shameless self-promotion: This is described in some detail in Chapter 9.5 of my book.
 
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WWGD

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Thanks! I had my first encounter with differential forms years ago in an inspiring class taught by Dave Metzler!



Differential forms allow one definition of volume and integration, even in the absence of a metric. Forms, and the wedge product for multiplying forms, are an abstraction of the concept of volume, which one can encounter in vector geometry as the area of a parallelepiped being the cross product of its sides.

Differential Geometry in Physics
Marián Fecko
http://davinci.fmph.uniba.sk/~fecko1/referaty/regensburg_2007.pdf
"Forms in L enter the play naturally when (oriented) volumes of parallelepipeds ... The tensor product α⊗β of two forms is not a form (just a tensor). If one, however, projects out the antisymmetric part of the result, one obtains a form."

Differential Forms and Integration
Terry Tao
https://www.math.ucla.edu/~tao/preprints/forms.pdf
"The concept of integration is of course fundamental in single-variable calculus. Actually, there are three concepts of integration ...

When one moves from single-variable calculus to several-variable calculus, though, these three concepts begin to diverge significantly from each other. The indefinite integral generalises to the notion of a solution to a differential equation ... The unsigned definite integral generalises to the Lebesgue integral, or more generally to integration on a measure space. Finally, the signed definite integral generalises to the integration of forms which will be our focus here. ...

The integration on forms concept ... yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds."
Nice description. I like the term used for the methodology of finding ( n-dimensional) volume using diff. forms : "Geometric Algebra". Basically the algebra somehow is parallel to the geometric constructions: a wedge of two basic form elements builds up an "area element" dx/\dy , a wedge of three basic forms dx/\dy/\dz builds up a volume element and so on. And these volume elements may be scaled for a given function by using the Jacobian.
 

Orodruin

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This discussion reminds me of one of my favourite differential forms, the symplectic 2-form ##\omega## on symplectic manifolds. It is used in Hamiltonian mechanics and I always felt uneasy with the concept of phase space volume until I understood that it was just the symplectic form to the nth power (in a 2n-dimensional symplectic manifold) and that the symplectic form is just given by ##\omega = dq^i\wedge dp_i##, where ##q^i## are generalised coordinates and ##p_i## the corresponding canonical momentum. This is a prime example of a situation where you do not have a metric to provide you with a volume form through the Levi-Civita symbol and the metric determinant.
 

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