Get the answer to "Let v=zk be the velocity field of a..."

High School
Geometry
High school geometry
Analytic geometry

midtlinjeg

Answered question

2021-02-25

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

Answer & Explanation

Szeteib

Skilled2021-02-26Added 102 answers

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

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