

Title: Proving Stokes' Theorem Post by Marissa on Apr 28^{th}, 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 zdirection are equal to the negative unit vector in the phidirection. 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 29^{th}, 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 semisphere with r = c, z >0, and v = rXz (v,r,z are vectors) The supposed solution is 2*pi*c^{2} 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 5^{th}, 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.gif_{C} 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.gif_{S} (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 xyplane. 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.gifu_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subtheta.gif}, where u_{http://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.gif_{C} ydx  xdy = 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifc^{2} = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_{S} 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 5^{th}, 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(r_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subvarphi.gif} x r_{http://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= 2c^{2} 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 xyplane. 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 xyplane, which is a circle of radius c. 

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