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riddles >> putnam exam (pure math) >> Proving Stokes' Theorem
(Message started by: Marissa on Apr 28th, 2008, 10:58am)

Title: Proving Stokes' Theorem
Post by Marissa on Apr 28th, 2008, 10:58am
How to prove stokes theorem (the integral of a vector over a closed trajectory = the integral of the rotational of the vector over a surface) for a particular problem where v = rXz where r is the vector of position (r*ur) and z a unit vector in the z direction (1*uz).

The vector v is supposed to equal r*sin(theta)* u_phi where u_phi is the unit vector in the phi direction. I understand that the cross product is equal to the magnitudes of the two components times the sine of the angle between them but I donīt understand why the cross product of the unit vector in the radial direction (which is supposed to be in spherical coordinates) and the unit vector in the z-direction are equal to the negative unit vector in the phi-direction.

I was told that the integral of the closed trajectory is equal to v*dl and that dl is equal to r*d(theta)*u_phi. I donīt understand this either. I think I get that the like is equal to the radius integrated around its closed trajectory but where does the unit vector come from? Another question on the same problem is, what are the bounds I am supposed to integrate about for the closed trajectory?

any thoughts, please.


// modified problem title to be more descriptive -- wwu

Title: Re: help with this problem
Post by Marissa on Apr 29th, 2008, 4:03am
The question is stated as follows: Verify Stokes theorem by calculating both members of
[img:111:27]http://www.postimage.org/aVITqX9.jpg[/img] (http://www.postimage.org/image.php?v=aVITqX9)
where S is a semi-sphere  with r = c, z >0, and v = rXz  (v,r,z are vectors)

The supposed solution is -2*pi*c2

The definition of Stokes theorem I have been given is (and I apologize if it doesn't sound quite right): The flux of the rotational of a field vector through a surface is equal to the circulation of the vectoral field along the line bounding the surface.

Any help would be greatly appreciated. Even just a push in the right direction

Title: Re: help with this problem
Post by Eigenray on May 5th, 2008, 6:37pm
Having just finished teaching a multivariable calc course, I will use the notation I am used to:

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/oint.gifC Fhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifdr = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifS (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifx F)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifndhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif.

Here F(r) = r x k, S is the upper hemisphere, and C is its boundary, a circle in the xy-plane.  Also, dr = Tds, where T is a unit tangent, and s is the arclength parameter, so that ds = |dr|, and n is a unit vector normal to the surface.  In Cartesian coordinates,

F = (y, -x, 0),
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifxF = (0, 0, -2).

In terms of spherical coordinates, F = -http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifuhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subtheta.gif, where uhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subtheta.gif = (-sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, 0) is the unit vector in the http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif direction.  I am using the coordinates

(x,y,z) = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif),

so I guess we have different notation for which of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif is which.

So the problem becomes

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/oint.gifC ydx - xdy = -2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifc2 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifS -2khttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifndhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif.

For the first integral, you just need to parameterize the curve C: x = c coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, y = c sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif,  0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif.

Title: Re: help with this problem
Post by Eigenray on May 5th, 2008, 6:37pm
To compute the second integral, there are a couple options.  Here are two:

1) Parameterize the surface.  Using spherical coordinates, r(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) = (c sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, c sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, c coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif), with 0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, 0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif.  Then

ndhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif= http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gif(rhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subvarphi.gif x rhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subtheta.gif),

and -2khttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifndhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif= -2c2 sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif, assuming we take n to be the outward (hence upwards) pointing normal.  We then integrate this over the parameterization, 0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, 0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif.

2) Think of the surface as the level surface g = 0, and use projection onto the xy-plane.  Then

n dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif= http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifg/ |http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifg http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifk| dxdy.

In this case, -2khttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gifndhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sigma.gif= -2dxdy, and we integrate over the projection onto the xy-plane, which is a circle of radius c.



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