Learning Stoke’s Theorem by example !

Calculus

Manmohan Dash<g6pontiac@gmail.com>

Question.

Calculate CurlF and then use Stokes’ theorem to compute the flux of CurlF through the given surface as a line integral. F  =   < y,  x,  x2 + y2 > , the upper hemisphere; x2 + y2 + z2 = 1,  z ≥ 0.

Answer; curlF =

i j k      
x y z
Fx Fy Fz

Now expanding the terms, the first component of curl of F is, (curlF)x = ∂yFz − ∂zFy and the 2nd term here has to be zero, as there is no z dependence. Thus the first term is 2y.

Similarly the y component of curl of F, is: (curlF)y = ∂xFz − ∂zFx = 2x.

But the 3rd term of the curl, (curlF)z = ∂xFy − ∂yFx is zero, since x-partial of 2nd component, ∂xFy, and y-partial of the 1st component of F, ∂yFx, are both 1.

So, CurlF =  < 2y,  2x,  0 > .

Now Stoke’s Theorem is the following; ∫S curlF.dS = ∮F.dl.

The line integral of F is over the contour of the circle on the z = 0 plane, as S on the surface integral on LHS is the upper hemisphere.

So, the dot product of F and dl is this: (yi + xj).(dxi + dyj) = ydx + xdy.

Consequently the integral is ∮ydx + xdy where the limits of integration vary from (0, 1).

So RHS = ∫01dx√1 - x2 + ∫01dy√1 − y2 = 2∫01dx√1 − x2.

We replace x = sinθ so that dx =  − cosθdθ.

We then have RHS =  − 2∫0 π/2dθcos2θ =  − 2[1/2(θ + sinθcosθ)]0 π/2 =  − π/2.

So the flux of the CurlF through the given surface is  − π/2.

Document generated by eLyXer 1.2.5 (2013-03-10) on 2015-05-30T22:42:34.204000

I am an experimental particle physicist, traveler, teacher, researcher, scientist and communicator of ideas. I am a quarkist and a bit quirky ! Hypothesis non fingo, eppur si muove, dubito cogito ergo sum are things that turn me on ! Researcher in experimental high energy physics (aka elementary particle physics; like “quarks, leptons & mesons and baryons”) … Teacher of Physics (and occasionally chemistry and maths) Blogger (check my website; mdashf.org) ! Love to read read and read but only stuff that interest me. Love to puff away my time in frivolities, just dreaming and may be thinking. Right now desperately trying to streamline myself.

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