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 hemispere (z&gt;0) of the sphere x2+y2+z2=1

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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 hemispere (z&amp;gt;0) of the sphere x2+y2+z2=1

Marcus Herman · Accepted Answer

Step 1 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 x2+y2+z2=1 Formula used: Volume =∫∫∫ (div F)dv Step 2 V=(0, 0, z) ÷, F=ddx(0)+ddy(0)+ddz(z) On solving the value, ÷ F=0+0+1=1 Step 3 Volume =∫∫∫ (÷ F)dv Where, dv=43πr3 x2+y2+z2=1=r2 Where, r=1 43πr2 Substituting the value of r, then the required value is 4π3

Natalie Yamamoto · Accepted Answer

Step 1 Let v=zk be the velocity a fluid in R3. We need to calculate the flow rate in m3s through the upper hemishere, s≥0, of the sphere x2+y2+z2=1 First, we need to parametrize the upper hemisphere surhere surface by using the spherical coordinates: G(ϕ, θ)=⟨x, y, z⟩ G(ϕ, θ)=⟨cos⁡ϕsin⁡θ, sin⁡ϕsin⁡θ, cos⁡ϕ⟩ for, 0≤ϕ≤π2, 0≤θ2π Step 2 Now we need to find the tangents Yϕ &amp;amp; Tθ and the pointing normal n: Tϕ=∂∂ϕ =⟨−sin⁡ϕsin⁡θ, cos⁡ϕsin⁡ϕ, 0⟩ Tθ=∂∂θ =⟨cos⁡ϕcos⁡θ, sin⁡ϕcos⁡ϕ, −cos⁡θ⟩ n(Tϕ×Tθ)=|ijk−sin⁡ϕsin⁡θcos⁡ϕsin⁡ϕ0cos⁡ϕcos⁡θsin⁡ϕcos⁡ϕ−cos⁡θ| n=(−cos⁡ϕsin2θ, −sin⁡ϕsin2θ, −cos⁡ϕsin⁡θ) Step 3 Since we got the donward n, we need to convert it in the upward: n=(cos⁡ϕsin2θ,sin⁡ϕsin2θ,cos⁡ϕsin⁡θ) n=sin⁡ϕ(cos⁡ϕsin⁡θ,sin⁡ϕsin⁡θ,cos⁡phu) We can also calculate v×n before inserting in the formula to simplify it: v×n=⟨0, 0, z⟩×sin⁡ϕ⟨cos⁡ϕsin⁡θ,sin⁡ϕsin⁡θ,cos⁡ϕ⟩ =⟨0, 0, cos⁡ϕ⟩×sin⁡ϕ⟨cos⁡ϕsin⁡θ,sin⁡ϕsin⁡θ,cos⁡ϕ⟩

Let v=zk be the velocity field (in meters per second)

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