Find the outward flux of F(x, y, z) = 4x i + 4y j + 4(z - 2)k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (3, 0, 0), (3, 1, 0), (3, 0, 1), and (3, 1, 1).
A) 12
B) 36
C) 0
D) 1
E) 4

Question

Quizplus · Accepted Answer

Using Gauss's Divergence Theorem, the outward flux of a vector field through a closed surface is equal to the integral of the divergence of the vector field over the enclosed volume. In this problem, the rectangular box is not closed, so we need to add 6 surface integrals of the vector field across each of the faces of the rectangular box. The divergence of the vector field F(x,y,z) is 4+4+4 = 12. The integral of the divergence over the enclosed volume is: ∭div(F) dV = ∭12 dV = 12V where V is the volume of the rectangular box. The 6 surface integrals can be computed as follows: - The integral over the face with vertices (0,0,0), (0,0,1), (0,1,0), and (0,1,1) is zero, since the vector field is parallel to this face and therefore dot product will be zero. - The integral over the face with vertices (3,0,0), (3,1,0), (3,0,1), and (3,1,1) is also zero, for the same reason as above. - The integral over the face with vertices (0,0,0), (0,1,0), (3,0,0), and (3,1,0) is ∫∫ F(x,y,0) · ndS = ∫∫ <0,0,0> · <-1,0,0> dA = 0 - The integral over the face with vertices (0,0,0), (0,0,1), (3,0,0), and (3,0,1) is ∫∫ F(x,0,z) · ndS = ∫∫ <4x,0,4(z-2)> · <0,-1,0> dA = 8 - The integral over the face with vertices (0,0,0), (0,1,0), (0,0,1), and (0,1,1) is ∫∫ F(0,y,z) · ndS = ∫∫ <0,4y,4(z-2)> · <-1,0,0> dA = 0 - The integral over the face with vertices (0,1,0), (0,1,1), (3,1,0), and (3,1,1) is ∫∫ F(x,1,z) · ndS = ∫∫ <4x,4,4(z-2)> · <0,1,0> dA = 8 Adding up all the surface integrals gives a total of 16. Therefore, by Gauss's Divergence Theorem, the outward flux of F across the rectangular box is: 12V + 16 where V is the volume of the box. The volume is easily computed as V = (3-0)(1-0)(1-0) = 3, so the total outward flux is (12)(3) + 16 = 52 Simplifying this expression, we get Answer: B

Find the Outward Flux of F(x, Y, Z) = 4x