Exam 19: Flux Integrals and Divergence

If $\vec { F}$ is a constant vector field and S₁ and S₂ are oriented rectangles with areas 3 and 15 respectively, both orientated with the vector $\overrightarrow { { i } } ,$ is it true that $\int _ { S _ { 2 } } \vec { F } \cdot \vec { d A } = 5 \oint _ { S _ { 1 } } \vec { F } \cdot \vec { d A }$ ?

Suppose S is a disk of radius 4 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward".Calculate the flux of ${ \vec { F } } = 4 \vec { i } + 5 \vec { j } - 5 \vec { k }$ through S.

Compute the flux integral of the vector field $\vec { G } = - 6 y \vec { i } + 2 x \vec { j } + 6 x y z \vec { k }$ through the square $0 \leq y \leq 2,0 \leq z \leq 2$ in the yz-plane, oriented so that the normal vector points in the direction of the x-axis.

Let S be the sphere of radius 4 centered at the origin, oriented outward. Find $\vec { n }$ , the unit normal vector to S in the direction of orientation.

Explain why it is impossible to give the Möbius strip a continuous orientation.

If $\vec{F}=-4 \vec{G}$ and S is an oriented surface, then $\int _ { S } \vec { F } \cdot \vec { d A } = - 4 \int _ { S } \vec { G } \cdot \vec { d A }$ .

Let ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$ be a constant vector field and S be an oriented surface. Show that $\int _ { S } \vec { F } \cdot \vec { d A } \leq \left( \sqrt { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } \right) ( \text { Area of } S )$

If $\int_{S} \vec{F} \cdot \overrightarrow{d A}=0$ , then $\vec{F}$ is perpendicular to the surface S at every point.

Let ${ \vec { F } } = y \vec { i } + x \vec { j }$ .Write down an iterated integral that computes the flux of $\vec { F }$ through S, where S is the part of the cylinder $x ^ { 2 } + y ^ { 2 } = 9$ with x $\ge$ 0, y $\ge$ 0, 0 $\le$ z $\le$ 4, bounded between the planes y = 0 and y = x, oriented outward.

Calculate the flux of ${ \vec { F } } = 5 \vec { i } + 2 \vec { j } - 2 x \vec { k }$ through the plane rectangle z= 1, 0 $\le$ x $\le$ 5, 0 $\le$ y $\le$ 2, oriented downward.

Calculate the flux of $\vec { F } = 5 \vec { i } + 5 \vec { j } - 2 x \vec { k }$ through the plane rectangle y = 1, 0 $\le$ x $\le$ 1, 0 $\le$ z $\le$ 3, oriented in the negative y-direction.

Compute the flux of the vector field ${ \vec { F } } = ( x - 4 y ) \vec { i } + ( 4 y - 2 z ) \vec { j } + 5 x \vec { k }$ 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.

If the surface area of surface S₁ is larger than the surface area of surface S₂, then $\int _ { S _ { 1 } } \vec { F } \cdot d \vec { A } \geq \int _ { S _ { 2 } } \vec { F } \cdot d \vec { A }$ .

Calculate the flux of $\vec{F}=3 \vec{i}+5 \vec{j}-6 \vec{k}$ through a surface of area 3 lying in the plane 2x + 5y + 5z = 10, oriented away from the origin.

Let $\vec { F } = \frac { x \vec { i } + y \vec { j } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } }$ . True or False: The flux of $\vec { F }$ through any cylinder C (with $x ^ { 2 } + y ^ { 2 } = R ^ { 2 } , 0 \leq z \leq 1$ )does not depend on the radius R.

Evaluate $\int _ { S } \vec { F } \cdot \vec { d A }$ , where ${ \vec { F } } = x \vec { i } + 6 y \vec { j }$ and S is the part of the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ with x $\ge$ 0, y $\le$ 0, 0 $\ge$ z $\le$ 2, oriented toward the z-axis.

A greenhouse is in the shape of the graph $z = 36 - x ^ { 2 } - y ^ { 2 }$ with the floor at z = 0.Suppose the temperature around the greenhouse is given by $T = 3 x ^ { 2 } + 3 y ^ { 2 } + ( z - 6 ) ^ { 2 }$ .Let ${ \vec { H } } = - \nabla T$ be the heat flux density field. Calculate the total heat flux outward across the boundary wall of the greenhouse.

Let $\vec { F } = - 2 \vec { i } - 5 \vec { j } + 4 \vec { k }$ , and let S₁ be a horizontal rectangle with corners at (0,0,1), (0,2,1), (3,0,1)and (3,2,1), oriented upward; S₂ a rectangle parallel to the xz-plane, with corners at (1,3,1), (2,3,1), (1,3,5)and (2,3,5), oriented in the positive y-direction, and S₃ a rectangle parallel to the yz-plane, with corners at (1,2,1), (1,4,1), (1,2,5)and (1,4,5), oriented in the negative x-direction. Arrange $\int _ { S _ { 1 } } \vec { F } \cdot \overrightarrow { d A }$ , $\int_{S_{2}} \vec{F} \cdot \overrightarrow{d A}$ and $\int _ { S _ { 3 } } \vec { F } \cdot \overrightarrow { d A }$ in ascending order.

Calculate the flux of ${ \vec { F } } = \left( x ^ { 2 } + z ^ { 2 } \right) y \vec { j }$ through the disk $x ^ { 2 } + z ^ { 2 } \leq 25$ on the plane $y = 5$ , oriented in the positive y-direction.

Calculate the flux of ${ \vec { F } } = \left( x ^ { 2 } + z ^ { 2 } \right) y \vec { j }$ , through the plane rectangle y = 4, 0 $\le$ x $\le$ 4, 0 $\le$ z $\le$ 5, oriented in the positive y-direction.