# 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{of the sphere} x^{2} + y^{2} + z^{2} = 1.

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 .
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Szeteib
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: $\text{Volume}=\int \int \int \left(÷F\right)dv$
$V=\left(0,0,z\right)$
$÷F=\frac{d}{dx}\left(0\right)+\frac{d}{dy}\left(0\right)+\frac{d}{dz}\left(z\right)$ On solving the value, $÷F=0+0+1=1$
$\text{Volume}=\int \int \int \left(÷F\right)dv$ 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 $\frac{4\pi }{3}$