Alyce Wilkinson

2021-01-08

Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate (in cubic meters per second) through the upper hemisphere $(z>0)\text{}\text{of the sphere}\text{}{x}^{2}+{y}^{2}+{z}^{2}=1$ .

Szeteib

Skilled2021-01-09Added 102 answers

Similar to circle sphere is a two dimensional space where the set of points that are at the same distance r from a given point in a three dimensional space. In analytical geometry with a center and radius is the locus of all points is called sphere
Given:
The upper hemisphere of the sphere is ${x}^{2}+{y}^{2}+{z}^{2}=1$
Formula used:
$\text{Volume}=\int \int \int (\xf7F)dv$

$V=(0,0,z)$

$\xf7F=\frac{d}{dx}(0)+\frac{d}{dy}(0)+\frac{d}{dz}(z)$
On solving the value,
$\xf7F=0+0+1=1$

$\text{Volume}=\int \int \int (\xf7F)dv$
Where,
$dv=\frac{4}{3}\pi {r}^{3}$

${x}^{2}+{y}^{2}+{z}^{2}=1={r}^{2}$
Where, r = 1
$\frac{4}{3}\pi {r}^{3}$
Substituting the value of r,
then the required value is $\frac{4\pi}{3}$

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