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Let C be the unit circle in the xy-plane, oriented counterclockwise as seen from above. The divergence of the vector field F~ = (z, x, y) is zero, and as a result the flux through every surface with boundary C should be the same. Confirm that this is the case with the upper half of the unit sphere, the lower half of the unit sphere, and the unit disk in the xy-plane

Answers

Answer 1

Answer:

Upper half of the unit sphere (call it $S_1$ ): parameterize by

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

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

$(\partial\vec s)/(\partial v)*(\partial\vec s)/(\partial u)=(\cos u\sin^2v,\sin u\sin^2v,\cos v\sin v)$

Then the flux of $\vec F$ over this surface is

$\displaystyle\iint_(S_1)\vec F\cdot\mathrm d\vec S=\int_0^(\pi/2)\int_0^(2\pi)(\cos v,\cos u\sin v,\sin u\sin v)\cdot(\cos u\sin^2v,\sin u\sin^2v,\cos v\sin v)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^(\pi/2)\int_0^(2\pi)\cos u\sin^2v\cos v+\cos u\sin u\sin^3v+\sin u\cos v\sin^2v=\boxed{0}$

Lower half of the sphere (call it $S_2$ ): all the details remain the same as above, but with $\frac\pi2\le v\le\pi$ . The flux is again $\boxed{0}$ .

Unit disk (call it $D$ ): parameterize the disk by

$\vec s(u,v)=(u\cos v,u\sin v,0)$

with $0\le u\le1$ and $0\le v\le2\pi$ . Take the normal vector to be

$(\partial\vec s)/(\partial u)*(\partial\vec s)/(\partial v)=(0,0,u)$

Then the flux across $D$ is

$\displaystyle\iint_D\vec F\cdot\mathrm d\vec S=\int_0^(2\pi)\int_0^1(0,u\cos v,u\sin v)\cdot(0,0,u)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^(2\pi)\int_0^1u^2\sin v\,\mathrm du\,\mathrm dv=\boxed{0}$

Answer 2

Answer:

Final answer:

The flux through every surface with boundary C, such as the upper half of the unit sphere, the lower half of the unit sphere, and the unit disk in the xy-plane, should be the same and it is zero.

Explanation:

The divergence of the vector field F~ = (z, x, y) is zero. Therefore, the flux through every surface with boundary C, such as the upper half of the unit sphere, the lower half of the unit sphere, and the unit disk in the xy-plane, should be the same.

This can be confirmed by considering that the electric flux through a closed surface is zero if there are no sources of electric field inside the enclosed volume. Since there are no charges inside the surfaces mentioned, the flux through each surface is zero.

Therefore, the flux through the upper half of the unit sphere, the lower half of the unit sphere, and the unit disk in the xy-plane is the same, and it is zero.

Learn more about Electric Flux here:

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Related Questions

A study is given in which scientists examined data on mean sea surface temperatures (in degrees Celsius) and mean coral growth (in millimeters per year) over a several-year period at different locations. Here are the data: Sea Surface Temperature 29.69 29.87 30.16 30.22 30.48 30.65 30.80 Growth 2.63 2.57 2.67 2.50 2.47 2.38 2.25 Required:a. Find the mean and standard deviation of both sea surface temperature x and growth y and the correlation r between x and y. b. Find the least-squares line. The result should agree with your work in part (a) up to roundoff error. (Round your answers to three decimal places.)c. Say in words what the numerical value of the slope tells you.
Which one is it ?? Please help
Round 3,721 to the nearest hundred.
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The amount of time a passenger waits at an airport check-in counter is random variable with mean 10 minutes and standard deviation of 2 minutes. Suppose a random sample of 50 customers is observed. Calculate the probability that the average waiting time waiting in line for this sample is (a) less than 10 minutes (b) between 5 and 10 minutes

The probability of finding a broken cookie in a bag of chocolate chip cookies is P = .03. Find the probability of getting at least 2 broken cookies in a bag containing 36 cookies

Answers

P(at least 2 broken cookies) = 1 - P(X = 0) - P(X = 1)

How to find the probability of getting at least 2 broken cookies in a bag containing 36 cookies

To find the probability of getting at least 2 broken cookies in a bag containing 36 cookies, we need to calculate the probability of getting 2, 3, 4, ..., up to 36 broken cookies and then sum up those probabilities.

The probability of getting exactly 2 broken cookies can be calculated using the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Using the formula, we can calculate P(X = 2):

P(X = 2) = C(36, 2) * (0.03)^2 * (1 - 0.03)^(36 - 2)

Similarly, we can calculate P(X = 3), P(X = 4), and so on, up to P(X = 36).

Once we have calculated all these probabilities, we can sum them up to find the probability of getting at least 2 broken cookies:

P(at least 2 broken cookies) = P(X = 2) + P(X = 3) + P(X = 4) + ... + P(X = 36)

P(at least 2 broken cookies) = 1 - P(X = 0) - P(X = 1)

To calculate P(X = 0), we can use the binomial probability formula with k = 0, and for P(X = 1), we can use the formula with k = 1.

Once we have calculated P(X = 0) and P(X = 1), we can substitute them into the equation:

P(at least 2 broken cookies) = 1 - P(X = 0) - P(X = 1)

This will give us the probability of getting at least 2 broken cookies in a bag containing 36 cookies.

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the answer would be .06 bc its 2 broken cookies in a bag therefore it would be double so the answer is .06

Which of the following is equivalent to 2 2/5

Answers

Answer:

0.4 - 0.40 - 4/10 -

Step-by-step explanation:

Can you please help Solve: 4 + x/7 = 2

Answers are:

-14
10
12
42

Answers

X=-14 because I did it before

Find the measurement of the numbered angles

Answers

Answer:

m<1 = 60

m<2 = 30

m<3 = 80

Step-by-step explanation:

1. Solve for angle (1)

The sum of angles in any triangle is (180) degrees. As one can see, there is a (30) degree angle in this triangle, and a (90) degree angle. Bear in mind that the box around an angle indicates that it is a (90) degree angle. One can form an equation and solve for the unknown angle using this given information;

(30) + (m<1) + (90) = 180

Simplify,

120 + m<1 = 180

Inverse operations,

m<1 = 60

2. Solve for angle (2)

The vertical angles theorem states that when two lines intersect, the angles opposite each other are congruent. One can apply this theorem here by stating the following,

m<2 = 30

Thus one gets their answer, the measure of angle (2) must be (30) degrees by the vertical angles theorem.

3. Solve for angle (3)

As states above, the sum of angles in a triangle is (180) degrees. Since one has found the measure of angle (2), one can form an equation and solve for the measure of angle (3) using the given information, combined with the information found.

(m<2) + (70) + (m<3) = 180

Susbtitute,

30 + 70 + (m<3) = 180

Simplify,

100 + m<3 = 180

Invers eoperations,

m<3 = 80

A flagpole casts a shadow that is 30 feet long. At the same time, a man standing nearby who is 6 feet tall casts a shadow that is 60 inches long. How tall is the flagpole to the nearest foot?

Answers

Answer:

5 feet and 60 inch

Step-by-step explanation:

5 feet and 60 inches

Step-by-step explanation:

You spend $28$28 on ingredients to make cookies. You charge $4$4 per container of cookies. How many containers do you need to sell to earn $20 in profit?

Answers

Let, the number of Containers = x

Then, Equation would be: 4x - 28 = 20

4x = 20 + 28

x = 48/4

x = 12

In short, Your Answer would be: 20 Containers

Hope this helps!

4x7=28 you need 7 just to break even then
4x5=20 you need 5 to make $20
7+5=12 You need 12 to make $20

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