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.

Question
Analytic geometry
asked 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 \((z > 0)\ \text{of the sphere}\ x^{2} + y^{2} + z^{2} = 1\).

Answers (1)

2021-01-09
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: \(\text{Volume}= \int \int \int ( \div F)dv\)
\(V = (0, 0, z)\)
\(\div F=\frac{d}{dx}(0)+\frac{d}{dy}(0)+\frac{d}{dz}(z)\) On solving the value, \(\div F = 0 + 0 + 1 = 1\)
\(\text{Volume} = \int \int \int (\div F)dv\) 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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