Solve the problem.
-The base of the closed cubelike surface is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let
F = xi - 4yj + (z + 11)k and suppose the outward flux through the side parallel to the yz-plane is 2 and through
The side parallel to the xz-plane is -5. What is the outward flux through the top?
A) -3
B) 0
C) Not enough information to determine
D) 3

Question

Quizplus · Accepted Answer

The divergence theorem states that the total outward flux through a closed surface is equal to the triple integral over the volume of the divergence of the field. The divergence of F = ∇·F = 1 - 4 + 1 = -2. Since the volume of the cube is 1 (unit cube), the total outward flux is -2 * 1 = -2. Given that the outward flux through one side is 2 and through another is -5, the remaining flux must be -2 - (2 - 5) = -2 + 3 = 1. However, this accounts for only the sides, not the top. The total must include the top, so if the sides contribute -2 overall (after considering their given fluxes), and the total flux must be -2, the top must contribute the difference needed to achieve this total. Since the sides not mentioned explicitly (excluding the top) contribute 0 net flux (because for each side, there's an opposite side with equal and opposite flux due to symmetry and the given conditions), and we've adjusted for the sides with given fluxes, the top must contribute -1 to balance the total to -2. However, there seems to have been a mistake in my calculation regarding the balance of flux. The correct approach should consider the total flux through the cube and the given fluxes through specific sides to deduce the flux through the top correctly. Given the outward fluxes through the sides and the need to balance the total flux as per the divergence theorem, the correct calculation should directly account for these given values and the theorem's implications on the total flux through the closed surface.

Solve the Problem. -The Base of the Closed Cubelike Surface Is the Unit