Please help I have no clue what I'm doing!!! Thanks :)

Let f(x,y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that li

Sophie [7]

Consider the closed region $V$ bounded simultaneously by the paraboloid and plane, jointly denoted $S$ . By the divergence theorem,

$\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

And since we have

$\nabla\cdot\mathbf f(x,y,z)=1$

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have

$\displaystyle\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\iiint_V\mathrm dV$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=2}^{z=3-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=1}r(3-r^2-2)\,\mathrm dr$
$=\dfrac\pi2$

Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by $D$ , we have

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-\iint_D\mathbf f\cdot\mathrm dS$

Parameterize $D$ by

$\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+2\,\mathbf k$
$\implies\mathbf s_u\times\mathbf s_v=u\,\mathbf k$

which would give a unit normal vector of $\mathbf k$ . However, the divergence theorem requires that the closed surface $S$ be oriented with outward-pointing normal vectors, which means we should instead use $\mathbf s_v\times\mathbf s_u=-u\,\mathbf k$ .

Now,

$\displaystyle\iint_D\mathbf f\cdot\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(-u\,\mathbf k)\,\mathrm dv\,\mathrm du$
$=\displaystyle-4\pi\int_{u=0}^{u=1}u\,\mathrm du$
$=-2\pi$

So, the flux over the paraboloid alone is

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-(-2\pi)=\dfrac{5\pi}2$

6 0

3 years ago

AA3+2=AAA CC6+6=CBB (AB|C) ->S57->E73->S47->E57->S43->W26->S18->?

Paladinen [302]

The question asks:
"Mark Atilius was expecting news from his friends with whom he agreed to reveal the great secret pyramids and spent his time at a nearby inn when he caught the attention of the Egyptian sitting beside him. He was even more surprised when he talked to him.
- You're Mark Atilius, are not you? she smiled - My name is Nefertari and I have a message for you from my grandmother. You should go right away if you want to get Pharaoh's belt you've been looking for all this time.
And he passed on the parchment he had just read.

 AA3 + 2 = AAA
CC6 + 6 = CBB
(AB | C) -> S57 -> E73-> S47-> E57-> S43-> W26-> S18->? 
Task: Find out the coordinates where Mark should come. "

First, you need to solve for the position from which Mark starts.
You know:
AA3 + 2 = AAA
Since 3 + 2 = 5,
553 + 2 = 555
Therefore A = 5.

Similarly:
CC6 + 6 = CBB
Since 6+6 = 12, B = 2.
In order from the middle digit to be 2, the original one must have been 1.
Therefore B = 2 and C = 1

Hence, the starting position is: (AB, C) = (52, 1)

The following line gives you how many steps and in what direction Mark should go: S = south (negative vertical motion), N = north (positive vertical motion), E = east (positive horizontal motion), W = west (negative horizontal motion).

(52, 1)
-> S57 -> (52, -56)
-> E73 -> (125, -56)
-> S47 -> (125, -103)
-> E57 -> (182, -103)
-> S43 -> (182, -146)
-> W26 -> (156, -146)
-> S18 -> (156, -164)

Hence, the coordinates that Mark should reach are (156, -164)

5 0

3 years ago

8p + 2q = -16 2p - q=2

yKpoI14uk [10]

Answer:

p = -1 q = -4

Step-by-step explanation:

a system of eq and solve for p and q ??? can do :)

Eq. 1) 8p + 2q = - 16

Eq. 2) 2p - q = 2

use Eq .2 and solve for q

2p - 2 = q

plug into Eq.1 with q

8p +2(2p - 2) = - 16

8p +4p -4 = -16

12p = - 12

p = -1

plug -1 into Eq. 1 for p and solve for q

8(-1) + 2q = - 16

-8 + 2q = - 16

2q = -8

q = -4

8 0

3 years ago

11 cups is equal to how many pints

dlinn [17]

It is 5.5 us pints.........hope it helps

8 0

3 years ago

Read 2 more answers

The distribution of income tax refunds follow an approximate normal distribution with a mean of $7010 and a standard deviation o

Andrej [43]

Answer:

A refund must be above $7,139 before it is audited.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

Approximately 68% of the measures are within 1 standard deviation of the mean.

Approximately 95% of the measures are within 2 standard deviations of the mean.

Approximately 99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 7010, standard deviation = 43.

Approximately 99.7% of the measures are within 3 standard deviations of the mean.

The empirical rule is symmetric, which means that the lowest (100-99.7)/2 = 0.15% is at least 3 standard deviations below the mean, and the upper 0.15% is at least 3 standard deviations above the mean.

Use the Empirical Rule to determine approximately above what dollar value must a refund be before it is audited.

3 standard deviations above the mean, so:

7010 + 3*43 = 7139.

A refund must be above $7,139 before it is audited.

5 0

2 years ago