Bachman's line integral versus classical line integral

In summary, David Bachman discusses the difference between "line integral" and "classical line integral" in his paper "A Geometric Approach to Differential Forms". He explains that the usefulness of linear differential forms is due to the Generalized Stokes theorem, which cannot be applied to non-linear forms. However, non-linear forms still have their uses, such as finding surface area over arbitrary surfaces. The 2-sphere in R3 is an example of a surface where a linear differential 2-form can be integrated to find its area, but Bachman also uses it as an example to demonstrate the need for non-linear forms to calculate surface area. In contrast, Lee mentions that differential forms are essential for use in Stokes' theorem but have the
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Rasalhague
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Bachman's "line integral" versus "classical line integral"

David Bachman A Geometric Approach to Differential Forms
http://arxiv.org/abs/math/0306194

When Bachman talks, in Appendix A, about "classical" line, surface, volume integrals, does he mean integrals of differential 0-forms (scalar fields) over 1-, 2- and 3-dimensional domains of R3. Is this the distinction he's making between the integrals of Appendix A (for which nonlinear differential forms are required) and the kind of line, surface, volume integrals he discussed in Chapter 5 (the kind to which Stokes' theorem applies), in which the integrand was a differential form of the same dimension as the domain of integration (for which linear differential forms suffice)?
 
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Bachman:

"The thing that makes (linear) differential forms so useful is the Generalized Stokes theorem. We don't have anything like this for non-linear forms, but that's not to say that they don't have their uses. For example, there is no differential 2-form on R3 that one can integrate over arbitrary surfaces to find their surface area."

In this thread, following quasar987's suggestion, I integrated a linear differential 2-form on R3 over a surface to find its area, didn't I? What is special about this surface that allowed me to find its area by integrating a linear differential 2-form, and why does Bachman use the same example, a 2-sphere embedded in R3, to demonstrate the need to integrate a nonlinear differential form if one wants to calculate surface area?!

Lee (Introduction to Smooth Manifolds):

Although differential forms are natural objects to integrate on manifolds, and are essential for use in Stokes' theorem, they have the disadvantage of requiring orientable manifolds.

The 2-sphere in R3 is orientable, isn't it? Likewise (the image of) a regular curve, and yet Bachman writes

Finally, we can define what is classically called a line integral as follows

[tex]\oint_C f(x,y) \; ds = \int_C f(x,y) \sqrt{dx^2+dy^2}[/tex]

(The circle an accident? He doesn't mention anything about loops in particular.) Both Bachman and Lee talk about line integrals as one of the motivating applications of linear differential 1-forms, but in these his concluding words, Bachman seems to be going back on that and redefining a line integral as a nonlinear differential form. A whole methodology was developed for pulling back k-forms from a k-dimensional domain in Rn to Rk where they can be treated as ordinary iterated integrals, a methodology which I thought was supposed to streamline and generalise "classical" line, surface, volume integrals; yet here, he seems to be saying that these methods can't be used for the very thing (line, surface, volume integrals) I thought they were designed for.

What is going on here?
 

1. What is the difference between Bachman's line integral and classical line integral?

Bachman's line integral and classical line integral are two different methods for evaluating line integrals in multivariable calculus. Bachman's line integral uses a different approach, known as the "Bachman kernel," which allows for more accurate calculations and can handle more complex integrands compared to the classical line integral.

2. When should I use Bachman's line integral instead of the classical line integral?

Bachman's line integral is typically used when the integrand is particularly complex or when the classical line integral fails to converge. However, it is always best to consult with your instructor or textbook to determine which method is most appropriate for a given problem.

3. Can you give an example of a problem where Bachman's line integral is used?

One example where Bachman's line integral is useful is when evaluating the work done by a force field along a curved path. The Bachman kernel allows for more precise calculations of this type of line integral compared to the classical method.

4. Are there any limitations to using Bachman's line integral?

As with any mathematical method, there are limitations to using Bachman's line integral. It may not always be applicable or efficient for certain types of problems, and it may also require a higher level of mathematical understanding compared to the classical method.

5. How do I learn to use Bachman's line integral?

If you are interested in learning how to use Bachman's line integral, it is recommended to consult with your instructor or refer to a reputable multivariable calculus textbook. There are also online resources and tutorials available that can help guide you through the process of using this method.

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