Science & Mathematics » Mathematics » F(x, y, z) = x^2 sin(z)i + y^2j + xyk, S is the part of the paraboloid z = 9 − x^2 − y^2 that lies above the xy-plane, oriented upward.?

# F(x, y, z) = x^2 sin(z)i + y^2j + xyk, S is the part of the paraboloid z = 9 − x^2 − y^2 that lies above the xy-plane, oriented upward.?

Use stoke's theorem

• curl((x^2)*sin(z),y^2, xy) = (x , (x^2)*cos(z) - y , 0)

vector(S) = (x , y , 9 - (x^2) - (y^2))

vector(dS) = (2x , 2y , 1) dy dx

(x , (x^2)*cos(z) - y , 0)·(2x , 2y , 1)

int_(-3)^(3) int_[-sqrt(9 - (x^2))]^[sqrt(9 - (x^2))] [2(x^2)*y*cos(z(x,y)) + 2(x^2) - 2*(y^2)] dy dx

int_(-3)^(3) int_[-sqrt(9 - (x^2))]^[sqrt(9 - (x^2))] [2(x^2)*y*cos(9 - (x^2) - (y^2)) + 2(x^2) - 2*(y^2)] dy dx

int_0^(2*pi) int_0^3 [2*((r*cos(theta))^2)*(r*sin(theta))*cos... - ((r*cos(theta))^2) - ((r*sin(theta))^2)) + 2*((r*cos(theta))^2) -
2*((r*sin(theta))^2)]*r dr dtheta

int_0^(2*pi) int_0^3 [(1/2)*((r^3)*sin(θ) + (r^3)*sin(3θ))*cos(9 - (r^2)) + 2*(r^2)*cos(2θ)]*r dr dθ = 0
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• Assuming that you want to compute ∫∫s curl F · dS:

Note that C is the curve x^2 + y^2 = 25 with z = 0.
Parameterize this by r(t) = <3 cos t, 3 sin t, 0> with t in [0, 2π].
Then, ∫∫s curl F · dS
= ∫c F · dr, by Stokes' Theorem
= ∫c <x^2 sin z, y^2, xy> · dr
= ∫(t = 0 to 2π) <(3 cos t)^2 sin 0, (3 sin t)^2, (3 cos t)(3 sin t)> · <-3 sin t, 3 cos t, 0> dt
= ∫(t = 0 to 2π) 27 sin^2(t) cos t dt
= 9 sin^3(t) {for t = 0 to 2π}
= 0.

I hope this helps!
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