A ray of light propagating in an isotropic medium with refractive index `n` varying gradually from point to points has a curvature radius `rho` determined by the formula
`(1)/(rho) = (del)/(del N) (In n)`,
where the derivative is taken with respect to the principle normal to the ray. Derivethis formula, assuming that is such a medium the law of refraction `n sin theta = ` const holds. Here `theta` is the angle between the ray and the direction of the vector `Delta n` at a given point.

Explanation

We consider a ray `QPR` in a medium of gradually varying refractive index `n`. At `P`, the gradient of `n` is a vector with the given direction while is nearly the same at neighbouring points `Q, R`. The are length `QR` is `ds`. We apply Snell's formula `n sin theta =` constant where `theta` is to be measured from the direction `Deltan`. the refractive indices at `Q,R` whise mid
point is `P` are `eta += (1)/(2)|Delta eta| d theta cos theta`
so `(eta - (1)/(2) |Delta eta| d theta cos theta) (sin theta +(1)/(2) cos theta d theta)`
`= (eta + (1)/(2) |Deltan| d theta cos theta) (sin theta - (1)/(2) cos theta d theta)` or `n cos theta d theta = | Deltan| ds cos theta sin theta`
(We have used here sin `(theta +=(1)/(2) d theta) = sin theta += (1)/(2) cos theta d theta)`
Now using the definition of the radius of curvature `(1)/(rho) = (d theta)/(ds)`
`(1)/(rho) = (1)/(eta) |Delta n| sin theta`
The equanity `|Delta n| sin theta` can be called `(deln)/(delN)` i.e. the derivation of `n` along the normal `N` to
the ray. Then `(1)/(rho) = (del)/(delN)`Inn.