Question

Let R be delimited by $x^2+y^2+z^2\leq 1$ and $z \leq 0$

and S being surface R, outwardly. Now give us the vector field $F: \mathbb{R}^3\rightarrow \mathbb{R}^3$

F(x,y,z)=ij +

calculate flux integral

Answer 1

Given F = (x + sin(z^2))i + (y^2 + cos(x^3)) j + (z^2 + sin(xy)) k Here, S = x^2 + y^2 + z^2 lessthanorequalto 1, x lessthanorequalto 0 In spherical coordinates, x = rho sin phi cos theta, y = rho sin phi sin theta, z = cos phi, Also, x^2 + y^2 + z^2 = p^2. Here, rho = squareroot 1 = 1. So, 0 lessthanorequalto rho lessthanorequalto 1, 0 lessthanorequalto theta lessthanorequalto 2 pi. Z lessthanorequalto 0 Rightarrow cos phi lessthanorequalto 0 Rightarrow phi lessthanorequalto pi/2 div F = partial differential P/partial differential x + partial differential Q/partial differential y + partial differential R/partial differential z = 1 + 2 rho sin phi sin theta + 2 rho cos phi then by using the divergence theorem: integral integral_s F . dS = integral integral integral_Q div F dV = integral^pi/2_0 integral^2 pi_0 integral^1_0