Q 6 - Calculus

Question

For what values of$a\in \left[0,\pi \right]$ in the spherical coordinate

system is the graph $\phi =a$ not a cone?

Step-by-Step Solution

Answer

The required answer is$\varphi =\alpha =\frac{\pi }{2}$ not a cone.

Step 1: Analyze phi = a

In spherical coordinates, \( \phi = a \) is a cone (half-angle \( a \)) for most values of \( a \). However, \( \phi = 0 \) is the positive z-axis (a ray, not a cone), and \( \phi = \pi \) is the negative z-axis. Also \( \phi = \pi/2 \) is the xy-plane (not a cone).

Step 2: Answer

For \( a \in (0, \pi) \), the graph \( \phi = a \) is not a cone when \( a = \frac{\pi}{2} \) (it is the xy-plane).