Write which of the following is a linear

Problem 4: Let F = (2z + 2)k be the flow field. Answer the following to verify the divergence theorem: a) Use definition to find

Viktor [21]

Given that you mention the divergence theorem, and that part (b) is asking you to find the downward flux through the disk $x^2+y^2\le3$ , I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere $x^2+y^2+z^2=3$ .

a. Let $C$ denote the hemispherical <u>c</u>ap $z=\sqrt{3-x^2-y^2}$ , parameterized by

$\vec r(u,v)=\sqrt3\cos u\sin v\,\vec\imath+\sqrt3\sin u\sin v\,\vec\jmath+\sqrt3\cos v\,\vec k$

with $0\le u\le2\pi$ and $0\le v\le\frac\pi2$ . Take the normal vector to $C$ to be

$\vec r_v\times\vec r_u=3\cos u\sin^2v\,\vec\imath+3\sin u\sin^2v\,\vec\jmath+3\sin v\cos v\,\vec k$

Then the upward flux of $\vec F=(2z+2)\,\vec k$ through $C$ is

$\displaystyle\iint_C\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^{\pi/2}((2\sqrt3\cos v+2)\,\vec k)\cdot(\vec r_v\times\vec r_u)\,\mathrm dv\,\mathrm du$

$\displaystyle=3\int_0^{2\pi}\int_0^{\pi/2}\sin2v(\sqrt3\cos v+1)\,\mathrm dv\,\mathrm du$

$=\boxed{2(3+2\sqrt3)\pi}$

b. Let $D$ be the disk that closes off the hemisphere $C$ , parameterized by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath$

with $0\le u\le\sqrt3$ and $0\le v\le2\pi$ . Take the normal to $D$ to be

$\vec s_v\times\vec s_u=-u\,\vec k$

Then the downward flux of $\vec F$ through $D$ is

$\displaystyle\int_0^{2\pi}\int_0^{\sqrt3}(2\,\vec k)\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv=-2\int_0^{2\pi}\int_0^{\sqrt3}u\,\mathrm du\,\mathrm dv$

$=\boxed{-6\pi}$

c. The net flux is then $\boxed{4\sqrt3\pi}$ .

d. By the divergence theorem, the flux of $\vec F$ across the closed hemisphere $H$ with boundary $C\cup D$ is equal to the integral of $\mathrm{div}\vec F$ over its interior:

$\displaystyle\iint_{C\cup D}\vec F\cdot\mathrm d\vec S=\iiint_H\mathrm{div}\vec F\,\mathrm dV$

We have

$\mathrm{div}\vec F=\dfrac{\partial(2z+2)}{\partial z}=2$

so the volume integral is

$2\displaystyle\iiint_H\mathrm dV$

which is 2 times the volume of the hemisphere $H$ , so that the net flux is $\boxed{4\sqrt3\pi}$ . Just to confirm, we could compute the integral in spherical coordinates:

$\displaystyle2\int_0^{\pi/2}\int_0^{2\pi}\int_0^{\sqrt3}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=4\sqrt3\pi$

4 0

3 years ago

I WILL GIVE YOU BRAINLIEST!!

ollegr [7]

Answer:

32 goldfish

Step-by-step explanation:

Given:

Number of fish tanks = 8
Volume of water per fish tank = 5,544 in³
Water required by 1 goldfish = min 1,386 in³

Number of fish per tank = water per tank ÷ water per goldfish

= 5544 ÷ 1386

= 4

Therefore, each tank can hold a maximum of 4 goldfish.

Greatest number of goldfish = number of tanks × fish per tank

= 8 × 4

= 32

3 0

2 years ago

Circumference of the cylinder a cylinder of the base with the diameter of 6cm and height of 12cm

yuradex [85]

Answer:

To find the volume, we must find the area of the circle and multiply it by its height, think of a cylinder as an object made up of MANY circles. So, V=pir^2h... so diameter is 6cm so radius = 3cm...

therefore, Volume = (pi)(3^2)(12) = 108(pi)

where 12 = height 3 = radius

For the surface area, we need to find the area of the 2 circles(top and bottom ) so once again,

2(pi)(r^2) = 2(pi)(9)

for the area of the rectangle(if we unfold the cylindrical can) we get D(pi)(12) where D= Diameter

so the surface area is

SA = (2)(9)(pi) + 6(pi)(12)

SA = 18pi + 72pi = 90pi

7 0

2 years ago

Read 2 more answers

What is the inverse of the statement:<br><br> If it is Saturday, then it is the weekend.

Harlamova29_29 [7]

It is the weekend, if it is Saturday. (?)

3 0

3 years ago

Find the value of x. Give your answer in simplest radical forms

REY [17]

Use the Pythagorean theorem:

$x^2=5^2+6^2\\\\x^2=25+36\\\\x^2=61\to x=\sqrt{61}$

8 0

3 years ago