help me please? check the answer. thank you.

Question

If F(x,y,z)=(-xy)i-(xz)j+(2yz)k and C is the closed triangular curve oriented counterclockwise around the plane x+z=3 from y=0 to y=4 in the first octant viewed from above, use Stoke's theorem to evaluate ∮_CF• dr.

The answer is ∫⁴₀∫³₀ 3-u dudv is it true?

Answer 1

First, are you sure there isn't a typo here? The plane x + z = 3 between y = 0 and y = 4 is a rectangle, not a triangle. If the rectangle is the correct shape of the surface, then the limits of integration (under Stokes' Theorem) will in fact be x = 0 to x = 3 and y = 0 to y = 4. Looks like you also did a "change of variables" with u and v. That was not necessary. Let's stick with x and y.

Inside the integral you need the "dot product" of the "curl of F" with "the normal vector" to the surface at any point. We assume here that r = < x , y , 3 - x > which basically gives you the coordinates in 3 dimensions at any point on the surface of the plane. The normal vector n = the cross product as follows:

"derivative of r with respect to x" X "derivative of r with respect to y" (The result is the normal vector n ).

It looks like you made a small error in the integrand. The curl F = < 2z + x, 0 , x - z > [You need to show work here] and n = < 1, 0, 1 > . Take the dot product and what do you get? Replace z with 3 - x and you have your integrand. Isn't it "3 + x"? You should take your time to verify everything I have said here.

Now integrate and you should arrive at a number for an answer. [Hint: The answer is between 50 and 60]