Use Stokes theorem to evaluate integralC F middot dr Assume

Use Stokes\' theorem to evaluate integral_C F middot dr. Assume C is oriented counterclockwise as viewed from above. F = (2z + x)i + (y - z)j + (x + y)k; C the triangle with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1)

Solution

By Stokes theorem, the line integral over C goes to a surface integral oover S

this surface integral is computed from the surface integral of Curl of F over the surface of the triangle (1,0,0), (0,1,0),(0,0,1).

Now curl of F is ( d/dy(x+y) - d/dz(y-z) ) i + ( d/dz(2z+x) -d/dx(x+y) )j + ( d/dx(y-z) -d/dy(2z+x))k

the components of the curl are then 2i, j, 0

The triangle can be stated as a plane x+y+z=1

hence z= 1-x-y

Parametrize over x,y with this substitution:

Take surface integral of ( 2i+j+0.k).(i,j,k)dA over the triangle in counterclockwise direction.

from 0 to 1-x and 0 to 1, where dA=dx.dy

The integral yields 3*(x-x^2/2) evaluated from from 0 to 1

result 3/2