How can you derive the expression for ds from a light ray's path?

[Fernando Rios]

Homework Statement

Find the path followed by a light ray if the index of refraction (in polar coordinates) is proportional to r^-2.

Relevant Equations

∫nds

This is an example from a textbook. They show the following:

∫nds = ∫r^-2*ds = ∫r^-2*√((dr)²+r²*(dθ)²)

How do you obtain the expression for ds?

 

[Abhishek11235]

The arc length in Cartesian coordinate is given by :

$ds^2 = dx^2 + dy^2$
Which is Pythagoras theorem for an infinitesimal part of curve.

The transformation function to polar coordinate has form:

$x= r cos\theta$
$y=r sin\theta$Substituting in above equation and manipulating gives desired result.

 

[Fernando Rios]

Got it! Thank you for your help.