so if let's say the question is to let F(x,y,z) = (0,-z,y), let C be the circle of radius a in the yz-plane oriented clockwise as viewed from the positive x-axis. Let S be the region enclosed by C, with positive side in the positive x-direction.
Evaluate directly ∫_C (F∙ds)
now the main of this problem and any integration problem I come across is that I can't seem to find the right parametrisation for it. is there any shortcut per se to find a suitable parametrisation for any question?
Evaluate directly ∫_C (F∙ds)
now the main of this problem and any integration problem I come across is that I can't seem to find the right parametrisation for it. is there any shortcut per se to find a suitable parametrisation for any question?
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Parameterize C via polar coordinates:
x = 0, y = a cos t, z = a sin t with t in [0, 2π].
So, ∫c F ∙ dr
= ∫c <0, -z, y> ∙ dr
= ∫(t = 0 to 2π) <0, -a sin t, a cos t> ∙ <0, -a sin t, a cos t> dt
= ∫(t = 0 to 2π) a^2 dt, since cos^2(t) + sin^2(t) = 1
= 2πa^2.
I hope this helps!
x = 0, y = a cos t, z = a sin t with t in [0, 2π].
So, ∫c F ∙ dr
= ∫c <0, -z, y> ∙ dr
= ∫(t = 0 to 2π) <0, -a sin t, a cos t> ∙ <0, -a sin t, a cos t> dt
= ∫(t = 0 to 2π) a^2 dt, since cos^2(t) + sin^2(t) = 1
= 2πa^2.
I hope this helps!
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We use polar, because this is a circle.
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