Let v = $azk 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 x2 + y2 + z2 = 1.
Recall how to find flow rate through a surface, S, for a fluid with velocity vector field v. Choose the correct integral below to calculate the flow rate across the S (volume per unit time).
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To calculate the integral, you will also need to recall how to find a vector surface integral. Let F be a vector field. Let G(u, v) be an oriented parametrization of an oriented surface S with parameter domain D. Assume that G is one-to-one, and regular, except possibly at points on the boundary of D. Then we can calculate the vector surface integral as follows.
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If the orientation of S is reversed, the surface integral changes sign.
Before we can make any calculations, we must find an appropriate parametrization for the upper hemisphere of the sphere x2 + y2 + z2 = 1 since that is the surface through which the fluid is moving. Since this is part of a sphere, it will be easiest to convert the surface to spherical coordinates.
State the standard conversions.
x = R cos θnLR7UAX3BCJm0GaV3u/72JeiwW1gSAHaCOwajUByigB8KPSTQUucRaUVB86FjKm7PPBNVaBbWFI=
y = R sin θnLR7UAX3BCJm0GaV3u/72JeiwW1gSAHaCOwajUByigB8KPSTQUucRaUVB86FjKm7PPBNVaBbWFI=
z = RR3PdR02UtII8K4OjaBYIkqbkHYCqtMBiMOIod16PVXR2hQJFhe0ZuDwnMOOds1m8CZnqn92PTUU=
Because the surface through which the fluid is moving is only the upper hemisphere, we have the following bounds on the parameters θ and φ.
0 ≤ θ <
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r+4OUE9RiIO+fvD2CLJYcM3JGGvOnul8gh6cG6pLmbzuxCVrfIARG6fBeFKucXOwjX+lURsN80a8Ri/cnew14e2V3BJn8ULBM5kdEXezvPMaeKtwmEAT651d0U8Hvicud2m3Qxl9zzAQr33ilnoWT+diCx0jxK33FCeB6Ir0JqlhcbrgUcXQy3UOQ4mIj95TGKhy0j1fEbEsrsKspO8JsCU2PwIYel+t2/jiHwRgij5dZSG+udlFsRy95k5lwU/Lea0QrsYDb7nIpjunRRyyvxOH3SFKyn57hqO0M2rkwgqapYjhNw1kwWLNFxm73Sxwfb5fNGwlHwyXyNyLl2bjglmcbhO7bWimSoHJukMOk4JjEzGfg4jS62Slyn8jFE0cX3JCGflhYHYKeCWx+p4Bte527shgbQRBGN8oYDwvH61HMxXmuoxxaRbJyogxEO4WQXS6D+xPn111jt1lwp8Warlyg1BshsV8iUoE8aV0CxfbiE/yEwjsDAraQEw9z24ofiQ4WxAp6AhfLJn9LgrAmKeNtOeCc+bu0DoVejb7YaI/mO4mzNzrpTeV287ZbGe7So9Ju/atR4rWJmf9+Hq7WaG1XFaYSQZtaKyEs2FRlfc0GwOX/ITSool5twIurmlbZGdDlCOMtnYpeOU1cHZT0Xinr5R+r9Dc4Lqt1XRxoT6n4lzRVq0Xr2/nDIHfGb/Fu7MvAyz9Um0Rf88vtvFuGlNzndeTlu789e4TnR53+lVAG9mGo5LpLGAmVu02Qnl4bZs6qU6D+hro90YCXvFmYmjYV0BgPE89qYv+M7moZ/6stCUYkN+ImYz4ksdcLOxPOeLzNFgdbr2GVToizESGtrCy9Qyof9zMrbSuCagSPwF/uLz48Mpox/cSTXLSr34HwUQiznjll1etx4Vp8kie6gGyAKzfVwDeIhSjaFSitOUO6IddXeWhfwiRnEhkuTmExu1jp9fwNZXEzFmVACnKRR1KKM/rF+Ur9bkQMTDyVAjfbffKsi+W+A==Since the radius of the sphere is 1, we have the following parametrization.
G(θ, φ) = (cos θ sin φ, sin θ sin φ, cos φ)
Find Tφ and Tθ for G(θ, φ). Then use these vectors to compute n(φ, θ), the outward normal vector through the upper hemisphere.
Tφ = (∂G / ∂φ) =
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(φ, θ) = Tφ × Tθ =
sRTxkSaQ1rbzXfrdqgWCE16us9Nh8O3YC2HGAMoNj4yD58xsL8uLKoZdluGwZYi1+Y+70Qv9opQhZxwdmrXIdgWy+hz9Nmf8E4Pz/xTyV6kUHWk4wwe3S7nB+vryLYfQfyOx7VXQ0beWcxSjVOdkxUCRd7lZMlkH4XI4JU92KIfcw5kI+n8povgPZCPAcLlHYktGxF7QNwc3ktihMO3bITOcO/IlIyJoOvoVCSHWU38g9+rPVBLrQuIi5T/r71rpgwmFnX7klGi4CvLRm2B/8M1IVopVxuJ3lvN7L2byVzZvHR9M7N5+YlG8J1jJJ4I+gyIuSOcUh6KUeU4wgFInQOxxPRr13/EBDXGllURkQOmvHLAPOLGtWQGBhgQ+9dDU9DXLIRQVOIrrVF09Cnzqai006sLDxSQ5K4AQcmbWzHlEFydX/Rj8doMF0O2PoMl2Pz6rCeSDP9X5PFMS4CWvrs7QT49vf7Wgtx9tx0UJmlA3dd8IueLMkZuKH1qA0f5nAbaBFS0v4q5RhzNd2TrYQ4ppoF+v7l9yOIgTllFcPbLh6Z6M/079VIxIgtVGSJwvptZAlcD3Gu8Hi9wrrpSWukHqy/pLJIh6JllHjh7YNlOgOkw/yYTMckwDE+juwzBm0brg0RF3m4G3DMXIHn4X0gWeJq8tpB7dL2ptFhTLTHT69V2uRHLRgBv8RY4QAb5EPhzxBlKNtHGgZFNFuPQx/xAhWBfx5orwCEh9HwQoPTJjxNOTud7C11Xobnlhp7/FBJ4zncJjxQNzfQoLcH/7y5esHTljxxNH6D+6NKfAiEGWAkQwyhOKswVIkda6o5gzLqNA4M6qU04A/ldxTHLAJlKktEl4guekKsoCE2lrBTkMIX4qdUlOF0y4wGXoBoy/9gMztqom4ahV+27BJnL9doZLHy4Su+B+JJbMxQvi+Xz96JB7jSbDjjmJEm74Gf6YkSQA/H8q2lFBs3qanmkJuweZV8MeQoPwOyBP85K1hqCXXq8oP6jBk+lUQTt1dGWiZV5QHSDExVKJX41evqdFIePFknj1ul1/oR+qom37euM7CGhrbFFjr5f+oVxOKN0gZLe60g9WnhjNVjDpu2ZZUfO3SmSMyoCY5JO5RURde9UHKkc0LT/2jSWKnkWubaU+MmLQgHsBVF+RL6EsExpress the velocity field v = $azk in terms of the parameters φ and θ. In other words, calculate v(G(θ, φ)).
v = $azk
= <0, 0, $az>
Therefore v(G(θ, φ)) = <1Wh3cvJ2xF4=, 1Wh3cvJ2xF4=, nc1ItEz0kR4=cosφ>
Compute the dot product v(G(θ, φ)) ·n(φ, θ).
v(G(θ, φ)) · n(φ, θ) = <0, 0, $a cos φ> · sin φ <cos θ sin φ, sin θ sin φ, cos φ>
= nc1ItEz0kR4=sin φ cos2φ
Integrate to calculate the flow rate through the upper hemisphere.
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