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3 years ago

Q-Whats is another name for plane M Can anyone explain?

Mathematics

1 answer:

Vaselesa [24]3 years ago

4 0

Plane xyn, 3 points determine a plane, and points xyn are on plane m

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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

If a skydiver jumps of an airplane and falls at 212 miles per hour, what is the skydiver’s velocity and speed?

mario62 [17]

Speed is simply some number with some unit of speed at the end while velocity is speed with a direction.

The skydiver's velocity is 212 miles per hour downward.

The skydiver's speed is 212 miles per hour

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3 years ago

4(x – 2 + y)<br> What the answer

dmitriy555 [2]

Answer:

4x -8 +4y

Step-by-step explanation:

Distribute

4(x – 2 + y)

4*x -4*2 +4*y

4x -8 +4y

6 0

3 years ago

Mia buys a milkshake that costs 5 dollars and a hamburger that costs 8 dollars. If sales tax is 9%, what is the

victus00 [196]

14 dollars and 17 cents

5 0

3 years ago

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PLS HELP!<br>0.99 is what percent of 9.90?

Leno4ka [110]

Answer:

10%

Step-by-step explanation:

first let's write out the problem

0.99 is what percent of 9.90 = 0.99=9.90*x

We need to what c is so we would do the opposite operation of multiplication, division so we get,

0.99/9.90= .1

now we turn .1 into percent and get out answer

0.1= 10%

3 0

3 years ago

Read 2 more answers