# Integration of differential forms

• I
• JonnyG
In summary, the conversation discusses the confusion surrounding the integration of differential forms on a smooth n-manifold. It explores the idea of integrating a 1-form over a curve on a manifold and how it can be understood from both an elementary calculus perspective and a differential forms perspective. The conversation also touches on the role of ##dt## in the integration process and how it relates to the concept of a covector. Ultimately, the conversation concludes that differential forms are meant to give meaning to the integration process, but it is important to also understand the role of ##dt## in this context.
JonnyG
I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example:

Let ##M## be a smooth n-manifold, let ##\gamma: [a,b] \rightarrow M## be a smooth curve in M, let ##\omega## be a smooth 1-form on M. Then by definition, $$\int_{\gamma} \omega = \int_{[a,b]} \gamma^{*} \omega$$

Choosing smooth coordinates ##(x^i)## on ##M##, we can write ##\omega = w_i dx^i##. So then by definition, $$\int_{[a,b]} \gamma^{*} \omega = \int_a^b (w_i \circ \gamma) d\gamma^i = \int_a^b (w_i \circ \gamma)\frac{d\gamma^i}{dt} dt$$

From an elementary calculus point of view, this makes sense to me. We take the value of ## (w_i \circ \gamma)\frac{d\gamma^i}{dt} ## at a point on some rectangle of some partition, multiply it by the length of the subrectangle (which is approximated by ##dt##), then we let the mesh of the partition tend to ##0##. The summation makes sense to me because it's a summation of real numbers (the product I just mentioned is a product of real numbers).

Now when I think about this from a differential forms perspective, well ## (w_i \circ \gamma)\frac{d\gamma^i}{dt} ## is a function that is being multiplied by the 1-form ##dt##. The value of ## (w_i \circ \gamma)\frac{d\gamma^i}{dt} ## at a point of a subrectangle is a real number, but the value of ##dt## at a point is a covector, not a real number. So what we really have is a summation of a product of a real number with a covector, which is itself a covector, so how can the result of the integral be a real number?

If I understand you correctly, you are asking about $$\displaystyle{\int_a^b \underbrace{\left(\omega\circ \gamma\right)\cdot \dfrac{d \gamma}{dt} }_{=:F(t)}\,dt}$$
and how length times height can be found in there, if we use the language of differential forms.
JonnyG said:
The value of ##F(t)## at a point of a subrectangle is a real number, but the value of ##dt## at a point is a covector, not a real number.
You evaluated ##F(t)## at ##t## in that description, but refused to do the same for the second factor ##dt##. A covector is a function which assigns a real number to a vector, i.e. a direction. So if you evaluate both, you get a product of real numbers summed up along the line from ##t=a## to ##t=b##.

I'm not sure whether I like this picture or call it far fetched. One could as well interpret the ##dt## as pure symbolic sign meant to describe what runs in ##\int_a^b##.

Under the keyword exact you can find a specific example here, where I think the multiplications are more transparent. It is function times function and finally summed up over the points (of evaluation).

fresh_42 said:
You evaluated ##F(t)## at ##t## in that description, but refused to do the same for the second factor ##dt##. A covector is a function which assigns a real number to a vector, i.e. a direction. So if you evaluate both, you get a product of real numbers summed up along the line from ##t=a## to ##t=b##.

##dt## is a covector field and evaluated at a point, gives you a covector, which then acts on a tangent vector. So if we are evaluating ##F \cdot dt## at a point ##t_0##, then we get ##F(t_0) dt_{t_0}##, which is a real number multiplied by a covector. I don't see how it makes sense to sum a bunch of covectors to obtain a real number. I hope that clears up my question.
fresh_42 said:
I'm not sure whether I like this picture or call it far fetched. One could as well interpret the ##dt## as pure symbolic sign meant to describe what runs in ##\int_a^b##

But then this is just the usual thing you're told in elementary calculus: the ##dt## tells you what variable to integrate over, so then it seems kind of pointless to have created the theory of differential forms. Differential forms are supposed to be giving meaning to the ##dt##, but with this viewpoint I am just circling back to "it's the variable to integrate over".

##dt## is constantly ##1## along the integration path from ##t=a## to ##t=b##.
JonnyG said:
But then this is just the usual thing you're told in elementary calculus: the ##dt## tells you what variable to integrate over, so then it seems kind of pointless to have created the theory of differential forms. Differential forms are supposed to be giving meaning to the ##dt##, but with this viewpoint I am just circling back to "it's the variable to integrate over".
Sometimes a cigar is just a cigar. Differential forms give a meaning: The differential form we actually integrate is ##\omega##. That means we have to look for an answer in ##\int_\Gamma \omega ##. As soon as we break it down to a path, we are back at school and ##dt \equiv 1## along this path.

fresh_42 said:
##dt## is constantly ##1## along the integration path from ##t=a## to ##t=b##.

Do you mean that ##dt## is constantly ##1## in coordinates ? If that's what you mean, I don't see how that resolves my misunderstanding: Evaluating ##F \cdot dt## at the point ##t = t_0## gives me the product of a real number with a covector.

JonnyG said:
But then this is just the usual thing you're told in elementary calculus: the ##dt## tells you what variable to integrate over, so then it seems kind of pointless to have created the theory of differential forms. Differential forms are supposed to be giving meaning to the ##dt##, but with this viewpoint I am just circling back to "it's the variable to integrate over".
People have defined differential forms as "the things you integrate", so don't be alarmed. I've attached an example from a book called "Applied Differential Geometry" by William Burke that might help make sense of why it works.

etotheipi

## 1. What is the purpose of integrating differential forms?

Integrating differential forms is used to calculate the total flux or flow of a vector field over a given region. It is also used to find the area, volume, or higher-dimensional analogues of these quantities.

## 2. How is integration of differential forms different from traditional integration?

Integration of differential forms involves integrating over a manifold, rather than a traditional interval or region in Euclidean space. It also takes into account the orientation and parametrization of the manifold, and can be done using different coordinate systems.

## 3. What are the different types of integrals used in integration of differential forms?

The two main types of integrals used in integration of differential forms are line integrals and surface integrals. Line integrals are used to integrate over curves or paths in a manifold, while surface integrals are used to integrate over surfaces in a manifold.

## 4. What is the relationship between integration of differential forms and Stokes' theorem?

Stokes' theorem is a fundamental theorem in the field of differential forms, which relates the integral of a differential form over a manifold to its boundary. It is often used to simplify calculations in integration of differential forms, and has many applications in physics and engineering.

## 5. Can integration of differential forms be extended to higher dimensions?

Yes, integration of differential forms can be extended to higher dimensions, such as integrating over 3D volumes or higher-dimensional manifolds. This is known as multilinear algebra and is an important tool in many areas of mathematics and physics.

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