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Question Number 168339 by MikeH last updated on 09/Apr/22
$$\mathrm{A}\:\mathrm{point}\:\mathrm{in}\:\mathrm{rectangular}\:\mathrm{coordinates}\: \\ $$$$\left({x},{y},{z}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{represented}\:\mathrm{in}\:\mathrm{spherical} \\ $$$$\mathrm{coordinates}\:\left({r},\theta,\varphi\right)\:\mathrm{by}: \\ $$$$\:{x}\:=\:{r}\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\:{y}\:=\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\: \\ $$$${z}\:=\:\mathrm{sin}\:\varphi,\:\mathrm{0}\:\leqslant\:\theta\:\leqslant\:\mathrm{2}\pi\:,\:\mathrm{0}\:\leqslant\:\varphi\:\leqslant\:\pi \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{Jacobian}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{transformation}\:\frac{\partial\left({x},{y},{z}\right)}{\partial\left({r},\theta,\varphi\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region} \\ $$$$\mathrm{delimited}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sphere}: \\ $$$$\:\:{S}\:=\:\left\{{x},{y},{z}\:\in\mathbb{R}^{\mathrm{3}} \:,\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:\leqslant\:{R}^{\mathrm{2}} ,\:{R}>\mathrm{0}\right\} \\ $$