$1\mathrm{.}1(20$ points$)$ Integrating over different paths

The following two paths go from an initial point $P$ of coordinates $(x_1,y_1)$ to a final point $Q$ of the coordinates $(x_2,y_2),$ with $x_1{x}_2$ and $y_1{y}_2.$

Path I: straight lines from $(x_1,y_1)(x_2,y_1)(x_2,y_2)$

Path II: straight lines from $(x_1,y_1)(x_1,y_2)(x_2,y_2)$

$($a$)[10]$ Evaluate the two integrals

${\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}\text{I}}{\overset{\phantom{}}{}}}(dx+dy)$ and ${\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}II}{\overset{\phantom{}}{}}}(dx+dy)$

There is a Path III that goes from point $P$ to point $Q$ directly in a straight line. Evaluate

${\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}\text{I}\text{I}\text{I}}{\overset{\phantom{}}{}}}(dx+dy)$

Show that these integrals depend only on the two end points and thus the result can be written as $u(Q)-u(P).$ Identify a function $u(x,y)$ for this case.

$($b$)[8]$ Evaluate the two integrals

${\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}\text{I}}{\overset{\phantom{}}{}}}dv={\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}\text{I}}{\overset{\phantom{}}{}}}x(dx+dy),$ and $,{\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}II}{\overset{\phantom{}}{}}}dv={\displaystyle \underset{\text{P}\text{a}\text{t}\text{h}II}{\overset{\phantom{}}{}}}x(dx+dy)$

Comment on the results in light of the existence of a function $v(x,y)$ that gives the integrals via its values at the two endpoints.

$($c$)[2]$ The analogy of $dv$ in part$($b$)$ is that of $dQ$ in thermodynamics. For the case of $dQ,$ there is a factor $\frac{1}{T}$ that makes $\frac{dQ}{T}$ an exact differential. Discuss how you could turn $dv$ in part$($b$)$ into an exact differential, in light of your answer in part$($a$).$