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A-point-in-rectangular-coordinates-x-y-z-can-be-represented-in-spherical-coordinates-r-by-x-r-sin-sin-y-sin-sin-z-sin-0-2pi-0-pi-a-Calculate-the-Jacobian-o




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}
$$\mathrm{A}\:\mathrm{point}\:\mathrm{in}\:\mathrm{rectangular}\:\mathrm{coordinates}\: \\ $$$$\left({x},{y},{z}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{represented}\:\mathrm{in}\:\mathrm{spherical} \\ $$$$\mathrm{coordinates}\:\left({r},\theta,\varphi\right)\:\mathrm{by}: \\ $$$$\:{x}\:=\:{r}\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\:{y}\:=\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\: \\ $$$${z}\:=\:\mathrm{sin}\:\varphi,\:\mathrm{0}\:\leqslant\:\theta\:\leqslant\:\mathrm{2}\pi\:,\:\mathrm{0}\:\leqslant\:\varphi\:\leqslant\:\pi \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{Jacobian}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{transformation}\:\frac{\partial\left({x},{y},{z}\right)}{\partial\left({r},\theta,\varphi\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region} \\ $$$$\mathrm{delimited}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sphere}: \\ $$$$\:\:{S}\:=\:\left\{{x},{y},{z}\:\in\mathbb{R}^{\mathrm{3}} \:,\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:\leqslant\:{R}^{\mathrm{2}} ,\:{R}>\mathrm{0}\right\} \\ $$

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