Question

a) Express the following surfaces in spherical coordinates
i) xz=3
ii) x2 + y2 - z2=1
b) Find the cylindrical coordinates of the points where the Cartesian coordinates are
i). (6,6,8)
ii) (√2,1,1)

Answer 1

a) Interrelation for Cartesian and spherical coordinates:

$x = \rho\sin\theta\cos\phi\\ y = \rho\sin\theta\sin\phi\\ z = \rho\cos\theta$

i) xz = 3 =>

$\rho\sin\theta\cos\phi*\rho\cos\theta=3$ =>

$\rho^2\sin(2\theta)\cos\phi = 6$

ii) $x^2 + y^2 - z^2=1$ =>

$(\rho\sin\theta\cos\phi)^2 + (\rho\sin\theta\sin\phi)^2 - (\rho\cos\theta)^2 = 1$ =>

$\rho^2(\sin^2\theta(\cos^2\phi + \sin^2\phi) - \cos^2\theta) = 1$ =>

$\rho^2(\sin^2\theta - \cos^2\theta) = 1$ =>

$\rho^2\cos(2\theta) = -1$

b) a) Interrelation for cylindrical and Cartesian coordinates:

$r = \sqrt{x^2+y^2}\\ \theta = \arctan{\cfrac{y}{x}}\\ z = z$

i) $(6,6,8)_{Cartesian}$ =>

$(\sqrt{6^2+6^2}, \arctan{\cfrac66, 8})_{cylindrical}$ =>

$(6\sqrt2, 45^\circ, 8)$

ii) $(\sqrt2, 1, 1)_{Cartesian}$ =>

$(\sqrt{(\sqrt2)^2+1^2}, \arctan{\cfrac1{\sqrt2}}, 1)_{cylindrical}$ =>

$(\sqrt3, 35.264^\circ, 1)$