Exam Question

lusta

I was very curious about number 7,

Prove the every polar azimuthal projection satisfies

$T_\phi \cdot T_\theta = 0$.

How can we show for $\textbf{every}$ azimuthal projection

mega_late

Check it 1,2,
recall that azimuthal projection satisfy the equation $T(\phi,\theta)=(r(\phi)cos(\theta),r(\phi)sin(\theta))$. Then $T_{\phi}=((r'(\phi)cos(\theta),r'(\phi)sin(\theta))$ and $T_{\theta}=((r(\phi)(-sin(\theta)),r(\phi)cos(\theta))$. Then $T_{\phi} \cdot T_{\theta} = ((r'(\phi)cos(\theta),r'(\phi)sin(\theta)) \cdot ((r(\phi)(-sin(\theta)),r(\phi)cos(\theta)) = (r'(\phi)cos(\theta))(r(\phi)(-sin(\theta))) + (r'(\phi)sin(\theta))(r(\phi)cos(\theta)) = -(r'(\phi)cos(\theta))(r(\phi)(sin(\theta))) + (r'(\phi)sin(\theta))(r(\phi)cos(\theta)) = 0$