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Multivariable CalculusEasyChange of Variables & the Jacobian

Change of Variables & the Jacobian · Easy problem

In spherical coordinates x=ρsin⁡ϕcos⁡θx = \rho\sin\phi\cos\theta, y=ρsin⁡ϕsin⁡θy = \rho\sin\phi\sin\theta, z=ρcos⁡ϕz = \rho\cos\phi, the Jacobian determinant is ∂(x,y,z)∂(ρ,ϕ,θ)=ρ2sin⁡ϕ\dfrac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} = \rho^2\sin\phi. Evaluate it at ρ=3\rho = 3, ϕ=π6\phi = \dfrac{\pi}{6}.