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3 years ago

13

Help me for a cookie ;)

2 answers:

Wewaii [24]3 years ago

7 0

Answer:

rotation

Step-by-step explanation:

GIVE ME MY COOKIE

mylen [45]3 years ago

3 0

This transformation is called a rotation:-D

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Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of

GenaCL600 [577]

Close off the hemisphere $S$ by attaching to it the disk $D$ of radius 3 centered at the origin in the plane $z=0$ . By the divergence theorem, we have

$\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV$

where $R$ is the interior of the joined surfaces $S\cup D$ .

Compute the divergence of $\vec F$ :

$\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2$

Compute the integral of the divergence over $R$ . Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

$\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}$

So the volume integral is

$\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5$

From this we need to subtract the contribution of

$\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S$

that is, the integral of $\vec F$ over the disk, oriented downward. Since $z=0$ in $D$ , we have

$\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k$

Parameterize $D$ by

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

where $0\le u\le 3$ and $0\le v\le2\pi$ . Take the normal vector to be

$\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k$

Then taking the dot product of $\vec F$ with the normal vector gives

$\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v$

So the contribution of integrating $\vec F$ over $D$ is

$\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4$

and the value of the integral we want is

(integral of divergence of <em>F</em>) - (integral over <em>D</em>) = integral over <em>S</em>

==> 486π/5 - (-81π/4) = 2349π/20

5 0

3 years ago

What is the absolute value of 8

levacccp [35]

Answer:

8

Step-by-step explanation

Because 8 is 8

4 0

3 years ago

A number is 4 hundred more than 14 2 . what is the number?

egoroff_w [7]

$n=number \\ \\ n=400+142 \\ \\ n=542$

7 0

3 years ago

A ferris wheel is 10 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position

Murrr4er [49]

Answer:

The equation is:

f (t) = 4 + 5 (1 - cos (2pi t / 2))

Step-by-step explanation:

with the previous exercise we look for the equation for h = f (t)

So the data we have are

Wheel diameter = 10m (wheel radius = 5m)

1 wheel gets 1 revolution in 2 minutes.

the beginning of a entry will be related to that f (0) = 4

our wish is that f (z) get at least 4 with an amplitude of 5 (this value determines the radius of the wheel) for 2 minutes

with this the particle f (t) is transformed into

f (t) = 4 + 5 (1 - cos (2pi t / 2))

We know that the maximum value of cos in t will be 0, 1 -cos has minutes, the result will be as follows:

f (t) = 4 + 5 (1 - cos (2pi t / 2))

5 0

3 years ago

Dane is 10 times Brian's age. The difference in their age is 27, how old is Brian

WARRIOR [948]

Brian is 3 years old.
Hope this helps!

3 0

3 years ago

Read 2 more answers