Answered: "Let v = zk be the velocity field..."

Alyce Wilkinson

Answered question

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

Answer & Explanation

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:

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

\frac{4}{3} π r^{3}

Substituting the value of r, then the required value is

\frac{4 π}{3}

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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.

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