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13 Nov 2019
Use this formula to derive the formula for triple integration in spherical EXAMPLE 4 coordinates SOLUTION Here the change of variables is given by x = Ï sin(q) cos(8) y = Ï sin(q) sin(θ) z = We compute the Jacobian as follows: sin(Ï) cos(8) = | sin(q) sin(θ) -p sin(Ï) sin(θ) Ï cos(Ï) cos(8) Ï sin(Ï) cos(8) Ï cos(q) sin(8) X, Y, Z a(p, θ, Ï) cos(p) -p sin(p) -p sin(q) sin(θ) sin(Ï) cos(8) Ï cos(q) cos(8) Ï cos(q) sin(8) -Ï sin(p) | sin(p) cos(9) sin(q) sin(θ) -p sin(Ï) sin(8) Ï sin(Ï) cos(8) = cos(q)(-p2 sin(q) cos(q) sin2(8)-p? sin(q) cos(q) cos"(9) -r sin(q)(p sin2(q) cos2(8) + Ï sin2(q) sin2(8)) -p? sin(q) cos2(q)- =-p? sin(q) Since 0 Ï Ï, we have sin(Ï)20. Therefore x, y, z)-|-|-p2 sin(Ï)I = a(p, θ, Ï) and this formula gives (x, y, z) dV = sin(Ï) cos(8), Ï sin(q) sin(8), p cos(q)1p2 sin(Ï) which is equivalent to the formula for triple integration in spherical coordinates
Use this formula to derive the formula for triple integration in spherical EXAMPLE 4 coordinates SOLUTION Here the change of variables is given by x = Ï sin(q) cos(8) y = Ï sin(q) sin(θ) z = We compute the Jacobian as follows: sin(Ï) cos(8) = | sin(q) sin(θ) -p sin(Ï) sin(θ) Ï cos(Ï) cos(8) Ï sin(Ï) cos(8) Ï cos(q) sin(8) X, Y, Z a(p, θ, Ï) cos(p) -p sin(p) -p sin(q) sin(θ) sin(Ï) cos(8) Ï cos(q) cos(8) Ï cos(q) sin(8) -Ï sin(p) | sin(p) cos(9) sin(q) sin(θ) -p sin(Ï) sin(8) Ï sin(Ï) cos(8) = cos(q)(-p2 sin(q) cos(q) sin2(8)-p? sin(q) cos(q) cos"(9) -r sin(q)(p sin2(q) cos2(8) + Ï sin2(q) sin2(8)) -p? sin(q) cos2(q)- =-p? sin(q) Since 0 Ï Ï, we have sin(Ï)20. Therefore x, y, z)-|-|-p2 sin(Ï)I = a(p, θ, Ï) and this formula gives (x, y, z) dV = sin(Ï) cos(8), Ï sin(q) sin(8), p cos(q)1p2 sin(Ï) which is equivalent to the formula for triple integration in spherical coordinates
15 Jul 2022
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Sixta KovacekLv2
8 Jun 2019
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