Use Stokes Theorem to evaluate dr where Fx y z 2zi 8x 3y

Use Stokes\' Theorem to evaluate dr , where F(x, y, z) = 2zi + (8x - 3y)j + (3x + y)k, and C is the boundary of the triangular region formed by (1, 0, 0), (0, 1, 0), and (0, 0, 2), traversed counterclockwise when viewed from above.

Solution

the given triangle is

(1,0,0) (0,1,0) (0,0,2)

which is nothing but the region of integration comes from projecting 2x + 2y + z=2 in the first octant onto the xy-plane, this region is bounded by 2x + 2y = 2 and the x,y axes.

By Stokes\' Theorem,
c F · dr
= s curl F · dS
= <1, -1, 8> · <-z_x, -z_y, 1> dA, using cartesian coordinates with z = 2 - 2x - 2y
= <1, -1, 8> · <2, 2, 1> dA
= 8 dA.

Since the region of integration comes from projecting 2x + 2y + z=2 in the first octant onto the xy-plane, this region is bounded by 2x + 2y = 2 and the x,y axes.

So, 8 dA
= (x = 0 to 1) (y = 0 to 1 - x) 8 dy dx
= (x = 0 to 1) (8 - 8x) dx
= 8 - 4
= 4.