Question

Express the following surfaces in spherical coordinates :

yz=2

Answer 1

Solution:

$yz=2\ ...(i)$

We know that

$y=r\sinθ\sinϕ \\ z=r\cos{\theta}$

Putting these values in (i),

$yz= r\sinθ\sinϕ\ r\cos{\theta} \\=r^2\sin{\theta}\cos{\theta}\cos{\phi} \\=\frac{1}{2}r^2\sin{2\theta}\cos{\phi}$

Now,

$yz=2 \\ \Rightarrow \frac{1}{2}r^2\sin{2\theta}\cos{\phi}=2 \\ \Rightarrow r^2\sin{2\theta}\cos{\phi}=4, \ r>0,\ \theta\in[0,\pi], \ \phi\in[0,2\pi]$