Question

Use Stokes' Thm to evaluate.

F(x,y,z)=(x^2)(e^yz) i +(y^2)(e^xz) j +(z^2)(e^xy) k
S is the hemisphere x^2 +y^2 +z^2 =4
z is greater than or equal to 0
oriented upward.

Answer 1

Do you want to compute the flux of the field through the hemisphere?

Answer 2

Take cross product of F and then take the double integral

Answer 3

The flux is equal to the work done by the field over the boundary.

Answer 4

take cross product of F That will give you a new vector

Call it V2

then use this equation to integrate

Answer 5

It is very easy to see that this flux is zero.
The field restricted to the plane z = 0 becomes
\[F(x,y)=\{x^2 , y^2, 0\}
\]
This field is conservative since it is the gradient of
\[
f(x,y)= \frac {x^3} 3 + \frac {y^3}3
\]
The work done by this field over a closed path as
\[x^2 + y^2=4
\]
is zero.

Answer 6

Of course in the above, we used Stokes's Theorem to conclude the the flux thru the hemisphere is equal to the work done by the field on the circle of center 0 and radius 2 in
the xy-plane.