$4\mathrm{.}10$ Consider the infinitesimal quantity

$($x$+$y$^(2))$dx$+$xdy

$($a$)$ Is this an exact differential?

$($b$)$ Evaluate the integral of this differential between the points $(1,1)$ and $(2,2)$ along three

straight$-$line paths:

i$(1,1)->(1,2)->(2,2)$

ii$(1,1)->(2,1)->(2,2)$

iii$(1,1)->(2,2)$

$($c$)$ Suppose that Eq$.(4\mathrm{.}75)$ is divided by x$^(2).$ Show that the resulting differential is exact, so

you can write

df$=($x$^(2)-$y$)/($x$^(2))$dx$+($dy$)/($x$)$

$($d$)$ Evaluate $\backslash$int df along the paths in part $($b$),$ and show that it is the same as what you would

obtain if you calculated f$(2,2)-$f$(1,1).$