What is refractive index? Deduce the formula

$\mathrm{\mu }=\frac{\sin \left({\displaystyle \frac{\mathrm{A}+{\mathrm{\delta }}_{\mathrm{m}}}{2}}\right)}{\sin \left({\displaystyle \frac{\mathrm{A}}{2}}\right)}$

 

for the refraction through a prism.

Prove that for a spherical refracting surface

 $\frac{\mathrm{\mu }}{\mathrm{v}}-\frac{1}{\mathrm{u}}=\frac{\mathrm{\mu }-1}{\mathrm{R}}$ where $\mu$= refractive index of the refracting surface and $\mathrm{R}$= radius of the spherical surface, $\mathrm{u},\mathrm{v}$ are the distances of the object and image respectively from the pole of the surface

Prove the formula : $\frac{1}{f}=\left(\mu -1\right)\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$ for curved refracting surfaces, the symbols having their usual significance.