91Ó°ÊÓ

Express in spherical polar coordinates: $$ \left(x^{2}+y^{2}\right) / z^{2} $$

Express in spherical polar coordinates: $$ 2 z^{2}-x^{2}-y^{2} $$

Express in spherical polar coordinates: (i) \(\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\) (ii) \(\frac{\partial}{\partial x}\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}\)

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=x^{2}+y^{2}+z^{2}, \quad\) V: the cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\)

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=x y^{2} z^{3}, \quad \mathrm{~V}:\) the box \(0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c\)

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=x^{2}, \quad \mathrm{~V}:\) the region \(1-y \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 2\)

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=e^{-(x+y+z)}, \quad \mathrm{V}:\) the infinite region \(x \geq 0, y \geq 0, z \geq 0\)

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=x^{2}+y^{2}+z^{2}, \quad \mathrm{~V}:\) the sphere of radius \(a\), centre at the origin

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: \(\rho=\frac{\sin ^{2} \theta \cos ^{2} \phi}{r}, \quad \mathrm{~V}:\) the sphere of radius \(a\), centre at the origin

Find the total mass of a mass distribution of density \(\rho\) in region \(\mathrm{V}\) of space: $$ \rho=r^{3} e^{-r} $$ V: all space