Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } = 4$ between $z = 0$ and $z = 3$ Then the flux of $\text { F }$ through S is

A)18?
B)20?
C)24?
D)26?
E)28?

Question

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } = 4$ between $z = 0$ and $z = 3$ Then the flux of $\text { F }$ through S is

A)18?
B)20?
C)24?
D)26?
E)28?

Quizplus · Accepted Answer

The flux of $\mathbf{F}$ through $S$ can be calculated using the divergence theorem, which states that the flux through a closed surface is equal to the triple integral over the volume enclosed by the surface of the divergence of $\mathbf{F}$. The divergence of $\mathbf{F}$ is given by $\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}x + \frac{\partial}{\partial y}y + \frac{\partial}{\partial z}2z = 1 + 1 + 2 = 4$.The volume $V$ enclosed by $S$ and the planes $z=0$ and $z=3$ is a cylinder of radius 2 and height 3. The volume of the cylinder is $V = \pi r^2 h = \pi (2)^2 (3) = 12\pi$.The flux through $S$ is then the integral of the divergence over this volume, which is $4 \times 12\pi = 48\pi$. Since $\pi \approx 3.14$, $48\pi \approx 48 \times 3.14 = 150.72$, which is not among the provided options. However, the correct approach to calculate the flux directly through the surface $S$ involves evaluating the surface integral of $\mathbf{F}$ over $S$, not just applying the divergence theorem in this straightforward manner without considering the specific geometry and limits of the integral.Given the misunderstanding in the initial explanation, let's correct the approach:The correct way to find the flux through $S$ involves setting up and evaluating the surface integral directly or applying the divergence theorem correctly by considering the actual limits and geometry of the problem. The divergence of $\mathbf{F}$ is indeed $4$, and the volume of the cylinder is correctly calculated as $12\pi$. Multiplying the divergence by the volume gives the total flux out of the surface, which should be $48\pi$. However, the mistake was in the interpretation of the options provided, which do not directly match this calculation due to a misstep in the explanation process.Re-evaluating the situation, the correct answer should reflect the calculation of the flux through the given surface by properly applying the principles of vector calculus, which might involve more nuanced considerations of the surface's orientation and the vector field's behavior across it. Given the options and the mistake in the explanation, the focus should be on understanding how to apply these principles rather than the incorrect numerical result provided.

Let F(x,y,z)=xi+yj+2zk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }F(x,y,z)=xi+yj+2zk

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$