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September 2nd, 2017, 06:43 AM   #1
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Stoke's theorem

I found two different solved examples of Stoke's theorem. And I'm kinda confused with the orientation. I tried googling it but still not clear. Can somebody explain?

1. Says that we have curve K that is oriented clockwise and we are looking at it from point T (0,0,10).
We parameterize it with r and phi.
It says that f(r)xf(phi) points upwards and the normal P downwards. So we have - in the front of the integral. How do we know that?

2.We have curve that is oriented clockwise and we are looking at it from T(0,0,0).
Here it says that both f(r)xf(phi) and P are pointing upwards. Here we have +. Again how do we know that?
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September 2nd, 2017, 06:48 AM   #2
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What are f(r) and f(phi)?
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September 2nd, 2017, 06:52 AM   #3
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Quote:
Originally Posted by MATHEMATICIAN View Post
What are f(r) and f(phi)?
The first one:
K is intersection of:
z = 1+x^2+y^2
z=5


Parametrization:

x = rcos(phi)
y = rsin(phi)
z =5

f(r,phi) = (rcos(phi), rsin(phi), 5)

f(r) and f(phi) are first derivatives.
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September 2nd, 2017, 07:07 AM   #4
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Quote:
Originally Posted by sarajoveska View Post
f(r) and f(phi) are first derivatives.
Partial derivatives? Of what?
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September 2nd, 2017, 07:23 AM   #5
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Originally Posted by MATHEMATICIAN View Post
Partial derivatives? Of what?
First we parameterize the curve and we get f(r,phi) = ( rcos(phi), rsin(phi),5) then partial derivatives with respect to r and phi. Then cross product.
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