Testing my knowledge of differential forms

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Discussion Overview

The discussion revolves around the calculation of a line integral of the differential form along a parametrized half-circle. Participants are examining the integration process and identifying errors in the computation of the integral.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the integral ##\int_C dx## should yield a net change of ##-2r## as they traverse the half-circle.
  • Another participant questions the presence of an extra factor of ##r## in the integral calculation.
  • A participant provides a detailed calculation of the integral, showing the steps leading to the result of ##-2r^2##.
  • There is a clarification regarding the limits of the parameterization for the half-circle, which is stated to be from ##0## to ##\pi##.
  • One participant points out a potential error in the calculation related to the value of ##\cos(0)##.
  • A later reply acknowledges the mistake in the calculation after the correction is highlighted.

Areas of Agreement / Disagreement

The discussion includes a mix of agreement and disagreement, particularly regarding the calculation of the integral and the interpretation of the results. Participants are actively refining their understanding but do not reach a consensus on the correct outcome.

Contextual Notes

Participants express uncertainty about the integration steps and the implications of the parametrization. The discussion highlights potential misinterpretations of trigonometric values and their impact on the integral's result.

JonnyG
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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?
 
Last edited:
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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##?
 
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: $$\int_C dx = \int_0^\pi -r \sin t dt$$
$$= \int_0^\pi r(-\sin t ) dt $$
$$= r(\cos \pi - cos 0) $$
$$ = r(-r - r) $$
$$= -2r^2$$
 
Isn't ##\cos(0)## just 1 though?
 
Office_Shredder said:
Isn't ##\cos(0)## just 1 though?

Wow I see my mistake now. Thanks
 

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