Exam 19: Flux Integrals and Divergence

Calculate the flux of ${ \vec { F } } = 4 \vec { i } + 2 \vec { j } - 5 x \vec { k }$ through a disk of radius 5 in the plane x = 3, oriented away from the origin.

Find the flux of ${ \vec { F } } = \left( x ^ { 2 } + z ^ { 2 } \right) \vec { j }$ through the disk of radius 5 in the xz-plane, centered at the origin, and oriented upward.Give an exact answer.

Let $\vec { F }$ be a constant vector field with ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$ , where a, b, c are constants satisfying the condition $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1$ .Let S be a surface lying on the plane x + 4y - 5z = 10 oriented upward. If the surface area of S is 10, what is the smallest possible value of $\int _ { S } \vec { F } \cdot \vec { d A }$ , and what are the corresponding values of a, b, c?

Let S be an oriented surface with surface area 6.Suppose $\vec{F}=a \vec{i}+b \vec{j}+c \vec{k}$ is a constant vector field with magnitude 3.If the angle between $\vec{F}$ and $\vec { n }$ is $\pi$ /6 at each point of the surface S, determine the value of the flux integral $\int_{S} \vec{F} \cdot \overrightarrow{d A}$ .

Calculate the flux of $\vec { F} = \left( x ^ { 2 } + z ^ { 2 } \right) y \vec { i }$ , through the plane rectangle z = 3, 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 5, oriented in the positive z-direction.

Let S be the cylinder $x ^ { 2 } + y ^ { 2 } = 4,0 \leq z \leq 6$ .Find Q if $\int \left( \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \vec { i } + \frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \vec { j } + \vec { k } \right) \cdot d \vec { A } = Q \pi$ .

Let $\vec { n }$ be the unit normal vector of S.If the angle between $\vec{F}$ and $\vec { n }$ is less than $\pi$ /2 at each point of the surface, then $\int _ { S } \vec { F } \cdot \overrightarrow { d A } \geq 0$

Suppose S is a disk of radius 2 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward".Write down an area vector for the surface S.

Let $\vec { G }$ be the constant vector field $a \vec { i } + b \vec { j } + c \vec { k }$ . Find a condition on a, b and c such that $\int_{S} \vec{F} \cdot \overrightarrow{d A}=0$ for any surface S lying on the plane -5x + 3y - 2z = 1.

Compute the flux of the vector field $\vec { F } = ( x - 4 y ) \vec { i } + ( 4 y - 2 z ) \vec { j } + 6 x \vec { k }$ 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 upward. What is the answer if the plane is oriented downward?

Let $\vec{F}=\alpha x \vec{i}+b y \vec{j}+c z \vec{k}$ where a, b and c are constants.Suppose that the flux of $\vec { F }$ through a surface of area 3 lying in the plane y = 3, oriented in the positive y-direction, is 45.Find the flux of $\vec { F}$ through a surface of area 4 lying in the plane y = 3, oriented in the negative y-direction.

Let $\vec { F } = x z \vec { i } + y z \vec { j } + 9 x ^ { 2 } y ^ { 2 } \vec { k }$ .Calculate the flux of $\vec { F }$ through the surface oriented upward and given by z = f(x, y)= xy, over the region in the xy-plane bounded by the curves $y = x ^ { 2 }$ and $y = x ^ { 3 }$ between the origin and the point (1, 1).