Testing my knowledge of differential forms

In summary, the conversation discusses the calculation of the integral of ##dx## over the positively oriented half-circle ##C## of radius ##r##. The individual believes that the integral should result in a net change of ##-2r##, but after performing the calculation, they obtain ##-2r^2## instead. Through further discussion and clarification, it is determined that the mistake was made in the calculation of the cosine function, resulting in an extra factor of ##r##.
  • #1
JonnyG
233
30
I am test my knowledge of differential forms and obviously I am missing something because I can't figure out where I am going wrong here:

Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of ##\int_C dx## should give the net change in ##x## as we travel from the beginning to the end of the circle, right? This change should be ##-2r## since ##C## is of radius ##r##. However, ## x = r \cos t \implies dx = -r \sin t dt \implies \int_C dx = \int_0^\pi -rsint dt = -2r^2 \neq -2r##.

Where am I going wrong?
 
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  • #2
JonnyG said:
##\int_0^\pi -rsint dt = -2r^2 \neq -2r##.
Where am I going wrong?

Check this again- where are you getting the extra factor of ##r##?
 
  • #3
Infrared said:
Check this again- where are you getting the extra factor of ##r##?

I made an edit - the half circle is parametrized for ## 0 \leq t \leq \pi##.

As for the extra factor of ##r##, this is my calculation: $$\int_C dx = \int_0^\pi -r \sin t dt$$
$$= \int_0^\pi r(-\sin t ) dt $$
$$= r(\cos \pi - cos 0) $$
$$ = r(-r - r) $$
$$= -2r^2$$
 
  • #4
Isn't ##\cos(0)## just 1 though?
 
  • #5
Office_Shredder said:
Isn't ##\cos(0)## just 1 though?

Wow I see my mistake now. Thanks
 

1. What are differential forms?

Differential forms are mathematical objects that are used to study the properties of smooth functions and manifolds in differential geometry. They are a generalization of the concept of a multivariable function, and can be used to represent geometric quantities such as vectors, lines, and surfaces.

2. How are differential forms different from other mathematical objects?

Differential forms are different from other mathematical objects such as vectors and matrices because they are defined in terms of the coordinate system of the underlying space, rather than in terms of a specific basis. This allows them to capture more intrinsic geometric properties of a space and make calculations more coordinate-independent.

3. What are some applications of differential forms?

Differential forms have many applications in mathematics and physics. They are used in differential geometry to study smooth manifolds, in physics to describe quantities such as electric and magnetic fields, and in engineering to solve problems in fluid dynamics and solid mechanics.

4. How are differential forms used in integration?

Differential forms are used in integration to generalize the concept of the integral of a function to higher dimensions and more general spaces. They allow for the integration of functions over curves, surfaces, and higher-dimensional manifolds, and can be used to solve problems in vector calculus and calculus of variations.

5. What are some resources for learning about differential forms?

There are many resources available for learning about differential forms, including textbooks, online courses, and video lectures. Some recommended resources include "Differential Forms and Applications" by Manfredo P. do Carmo, "Introduction to Differential Forms" by Claudio Arezzo, and the "Differential Forms" course on Coursera by Paul Renteln.

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