A point in rectangular coordinates (x,y,z) can be represented in spherical coordinates (r,θ,ϕ) by: x = r sin θ sin ϕ, y = sin θ sin ϕ, z = sin ϕ, 0 ≤ θ ≤ 2π , 0 ≤ ϕ ≤ π (a) Calculate the Jacobian of the transformation ((∂(x,y,z))/(∂(r,θ,ϕ))) (b) Calculate the volume of the region delimited by the sphere: S = {x,y,z ∈R^3 , x^2 +y^2 +z^2 ≤ R^2 , R>0}


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Question Number 168339 by MikeH last updated on 09/Apr/22
$A point in rectangular coordinates (x,y,z) can be represented in spherical coordinates (r,θ,ϕ) by: x = r sin θ sin ϕ, y = sin θ sin ϕ, z = sin ϕ, 0 ≤ θ ≤ 2π , 0 ≤ ϕ ≤ π (a) Calculate the Jacobian of the transformation ((∂(x,y,z))/(∂(r,θ,ϕ))) (b) Calculate the volume of the region delimited by the sphere: S = {x,y,z ∈R^3 , x^2 +y^2 +z^2 ≤ R^2 , R>0}$
$A point in rectangular coordinates$ $(x, y, z) can be represented in spherical$ $coordinates (r, θ, φ) by :$ $x = r \sin θ \sin φ, y = \sin θ \sin φ,$ $z = \sin φ, 0 ⩽ θ ⩽ 2 π, 0 ⩽ φ ⩽ π$ $(a) Calculate the Jacobian of the$ $transformation \frac{\partial (x, y, z)}{\partial (r, θ, φ)}$ $(b) Calculate the volume of the region$ $delimited by the sphere :$ $S = {x, y, z \in R^{3}, x^{2} + y^{2} + z^{2} ⩽ R^{2}, R > 0}$
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