# Let v = zk be the velocity field of a fluid in R^3. Calculate the flow rate through the upper hemisphere (z > 0) of the sphere x^2 + y^2 + z^2 = 1.

Question
Analytic geometry
Let v = zk be the velocity field of a fluid in $$\displaystyle{R}^{{3}}$$. Calculate the flow rate through the upper hemisphere (z > 0) of the sphere $$\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={1}.$$

2021-02-26

Formula used:
Volume $$\displaystyle=\int\int\int{\left(\div{F}\right)}{d}{v}$$
V=(0,0,z)
$$\displaystyle\div{F}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({0}\right)}+\frac{{d}}{{{\left.{d}{y}\right.}}}{\left({0}\right)}+\frac{{d}}{{{\left.{d}{z}\right.}}}{\left({z}\right)}$$
On solving the value,
div F=0+0+1=1
Volume $$\displaystyle=\int\int\int{\left(\div{F}\right)}{d}{v}$$
Where,
$$\displaystyle{d}{v}=\frac{{4}}{{3}}\pi{r}^{{3}}$$
$$\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={1}={r}^{{2}}$$
Where, r=1
Substituting the value of r, then the required value is (4pi)/3ZSK

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