Denote the cylindrical surface by $S$ , and its interior by $R$ . By the divergence theorem, the integral of $\vec v$ across $S$ (the outward flow of the fluid) is equal to the integral of the divergence of $\vec v$ over the space it contains, $R$ :

$\displaystyle\iint_S\vec v\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec v)\,\mathrm dV$

The given velocity vector has divergence

$\vec v=z\,\vec\imath+y^2\,\vec\jmath+x^2\,\vec k\implies\nabla\cdot\vec v=\dfrac{\partial(z)}{\partial x}+\dfrac{\partial(y^2)}{\partial y}+\dfrac{\partial(x^2)}{\partial z}=2y$

Then the total outward flow is

$\displaystyle\iiint_R2y\,\mathrm dV$

Converting to cylindrical coordinates gives the integral

$\displaystyle2\int_0^{2\pi}\int_0^3\int_0^4r^2\sin\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{0}$

6 0

3 years ago

Aboard is cut into 12 equal pieces. How many pieces together represent 3/4 of the board? Explain how you arrived at your answer?

Sphinxa [80]

Answer:

9 pieces represent 3/ 4 because

3/4 × 12 parts = 9 parts

mark me as brainliest

4 0

3 years ago

Read 2 more answers

How many 1/3 pounds serving in 4 2/3 pound of cheese

Marysya12 [62]

Change 4 and 2/3 to an improper fraction.
4 + 2/3 = 14/3
There are 14 of the 1/3 lb serving in 4 and 2/3 pounds of cheese.

3 0

3 years ago

A radio is being discounted 20% if the sale price of the radio is $51.44 what is the original price of the radio Radio is being

Grace [21]

Original price of the radio is $64.30

Sale price - $51.44
Discount rate - 20%

This means that $51.44 is the discounted price or is equivalent to 80% of the original price, since the 20% equivalent is deducted from the original price.

To get the original price, divide $51.44 by its corresponding percentage, 80%.

$51.44 / 80% = $64.30

To get the discount, multiply the original price by its discount rate

$64.30 x 20% = $12.86

To get the sales price, deduct the discount from the original price.

$64.30 - $12.86 = $51.44
100% - 20% = 80%

6 0

3 years ago