Use Stokes' Theorem to compute the surface integral?

Question

Use Stokes' Theorem to compute the surface integral where S is the portion of the tetrahedron bounded by x+y+2z=2 and the coordinate planes with z>0, n upward, F=<zy^4-y^2, y-x^3, z^2>. (Answer: -4/3)

Answer 1

Start by finding the parametric form of the curves that bound the surface.  The 3 corners of the tetrahedron occur with the intersection of each coordinate axis, so simply set 2 of the variables equal to zero to get the intercept.

P1 = (2,0,0)     P2 = (0,2,0)      P3 = (0,0,1)

Then using these 3 points, write equations for the 3 lines that connect them as follows.

C1 = <2-2t, 2t, 0>  

C2 = <0, 2-2t, t>

 C3 = <2t, 0 , 1-t>  (Note that when you plug in t = 0 you get the first point, and when you plug in t = 1 you get the second).  

Now by Stoke's theorem, the surface integral of curl F dot dS = line integral of F dot dr.  We will need to calculate 3 line integrals for each curve.  

Set r(t) = C1, and find F(r(t)) = <-rt^2, 2t-(2-2t)^3, 0>, dr = <-2, 2, 0>

Then F*dr = 8t^2 + 2*(2t - (2-2t)^3)

Int{0,1}(F*dr) = 8/3*t^3 + 2*(t^2 + 1/8(2-2t)^4) = 8/3 + 2 - 4 = 2/3.

Repeat this process with C2 and C3 and sum up the solutions to the 3 integrals and you will get your answer of -4/3.  (int of C2 will be -5/3 and int of C3 will be -1/3)