Exam 19: Flux Integrals and Divergence

What is the flux of the vector field $\overrightarrow { \vec { F } } = - 5 \vec { i } + 6 \vec { j } + 2 \vec { k }$ through a circle in the xy-plane of radius 2 oriented upward with center at the origin?

Suppose S is a disk of radius 2 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward". Calculate the flux of $\vec { G } = \left( e ^ { - 2 x y } z \sec y \right)$ through S.

Compute the flux of ${ \vec { F } } = x z ^ { 2 } \vec { i } + y z ^ { 2 } \vec { j } + 4 z ^ { 5 } \vec { k }$ through the cylindrical surface $x ^ { 2 } + y ^ { 2 } = 16,0 \leq z \leq 5$ oriented away from the z-axis.

Let C be the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ of fixed radius R with $\pi$ /3 $\le$ $\theta$ $\le$ 2 $\pi$ /3 and -a $\le$ z $\le$ a oriented outward for some positive number a.Let S be the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = R ^ { 2 }$ with $\pi$ /3 $\le$$\theta$$\le$ 2 $\pi$ /3 and $\pi$ /3 $\le$ $\varphi$ $\le$ 2 $\pi$ /3 oriented outward.Determine the value of a for which the flux of $\vec { r }$ through each of these surfaces is equal in magnitude but opposite in sign for any choice of R.

Let S be the sphere of radius 3 centered at the origin, oriented outward. Suppose $\vec{F}$ is normal to $\vec { n }$ at every point of S.Find the flux of $\vec{F}$ out of S.

Suppose the surface S is the part of the surface x = g(y, z), for points (y, z)belonging to a region R in the yz-plane.If S is oriented in the positive x-direction, what will be the formula for computing the flux of $\vec { F}$ through S?

Compute the flux of the vector field $\vec { F } = y \vec { i } + ( x + y ) \vec { j } + z \vec { k }$ through the surface S that is the part of the surface $z = x ^ { 2 } - y ^ { 2 }$ above the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ , oriented in the positive z-direction.

Let S be the sphere of radius 6 centered at the origin, oriented outward. Let $\vec { G }$ be a vector field such that $\vec { G } = 3 x \vec { i } + 3 y \vec { j } + 3 z \vec { k }$ at every point of S.Find the flux of $\vec { G }$ out of S.

Compute the flux of the vector field ${ \vec { F } } = ( x - 3 y ) \vec { i } + ( 4 y - 2 z ) \vec { j } + 4 x \vec { k }$ through the surface S, where S is the part of the plane z = x + 2y above the rectangle $0 \leq x \leq 2,0 \leq y \leq 3$ oriented downward.

If $\vec { F }$ is a constant vector field and S₁ and S₂ are oriented rectangles with areas 1 and 2 respectively, then $\int _ { S _ { 2 } } \vec { F } \cdot \vec { d A } = 2 \int _ { S _ { 1 } } \vec { F } \cdot \vec { d A }$ .

Suppose T is the triangle with vertices $( 1,0,0 ) , ( 0,2,0 )$ and $( 0,0,2 )$ oriented upward.Calculate the flux of ${ \vec { F } } = e ^ { 8 x + 4 y + 4 z \vec { i } } - 5 \vec { j }$ through T exactly, and then give an answer rounded to 3 decimal places.

(a)Compute the flux of the vector field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$ through S_a, the sphere of radius a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward. (b)Find $\lim _ { a \rightarrow 0 } \int _ { S _ { 0 } } \vec { F } \cdot d \vec { A }$ .

Find the flux of ${ \vec { F } } = x \vec { i } + y \vec { j } + 8 z \vec { k }$ over the sphere S_a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward, with a > 0.

Calculate $\int _ { C } ( 5 x \vec { i } + 3 y \vec { j } ) \cdot \vec { d A }$ where C is a cylinder of radius R with 0 $\le$ z $\le$ 1.

Let ${ \vec { F } } = x \vec { i } + y \vec { j }$ .Write down an iterated integral that computes the flux of $\vec { F }$ through S, where S is the part of the surface $z = x ^ { 2 } + y ^ { 2 }$ below the plane z = 16, oriented downward.

Suppose that S is the surface which is a portion of the graph of a smooth function $z = f ( x , y )$ over a region R in the xy-plane, oriented upward.Consider the vector field $\vec { F } = f _ { x } ( x , y ) \vec { i } + f _ { y } ( x , y ) \vec { j } + g ( x , y , z ) \vec { k }$ . Find $g ( x , y , z )$ so that $\int _ { S } { \vec { F } } \cdot d \vec { A } = 0$ .

Let S be the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ with x $\ge$ 0, y $\ge$ 0, z $\ge$ 0, oriented outward.Evaluate $\int _ { S } ( x z \vec { i } + 4 y z \vec { j } ) \cdot \vec { d A }$ .

A circular disk, S, of radius 2 and centered on an axis, is perpendicular to the y-axis at y = -6 with normal in the direction of decreasing y. Consider the vector field $\vec { F } = x \vec { i } + y \vec { j } + ( z + x ) \vec { k }$ .Is the flux integral $\int_{S} \vec{F} \cdot \overrightarrow{d A}$ positive, negative or zero?

$\int_{S} \vec{F} \cdot d \bar{A}$ is a vector.

Let S be the spherical region of radius R with $\pi$ /3 $\le$ $\varphi$ $\le$ 2 $\pi$ /3 and $\pi$ /3 $\le$$\theta$ $\le$ 2 $\pi$ /3.Find the value of R so that $\int _ { S } \vec { r } \cdot \vec { d A } = 40 \pi$ Give your answer to two decimal places.