Many flow phenomena can be understood if one learns to approximately visualize the flow pattern via streamlines of the flow. An important step in this direction is to represent the kinematics of the flow using streamline coordinates as shown in fig. $1.$ The figure shows a streamline in a $2$D flow $($for simplicity$)$ at a given instant $\text{'}t\text{'}.$ Any point $($say P $)$ on the streamline can be located with the help of a streamwise distance $\text{'}$s$\text{'}$ measured from some reference. Also the local normal $($from concave towards convex side$)$ coordinate at the point along with the corresponding unit vectors, hat$(e{)}_s$ and hat$(e{)}_n$ can be readily constructed as shown in fig $1.$ Let $$ be the local slope and $\text{'}$ R $\text{'}$ the local radius of curvature of the streamline at the point P$.$ 'Show that

$($a$)\frac{\text{d}\text{e}\text{l}\text{h}\text{a}\text{t}(e{)}_n}{\text{d}\text{e}\text{l}\text{t}}=+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}$hat$(e{)}_n,\frac{\text{d}\text{e}\text{l}\text{h}\text{a}\text{t}(e{)}_e}{\text{d}\text{e}\text{l}\text{s}}=-\frac{\text{h}\text{a}\text{t}(e{)}_n}{R}$

$($b$)$ Acceleration of the particle: vec$()=(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{t}}+V\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{s}})hat(e{)}_s+(V\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}-\frac{V^2}{R})hat(e{)}_n$; where V is the instantaneous speed of the fluid particle at point P on the streamline

$($c$)$ Simplify the answer obtained in $($b$)$ for a steady flow

$($d$)$ Simplify the answer obtained in $($b$)$ for a steady flow with straight streamlines