Chapter 3, Problem 3.24NP

Interpretation Introduction

Interpretation: The differential ${\text{dz = 2xydx + x}}^2\text{dy}$ is to be shown as exact by integrating $\text{dz}$ along the following paths:

  $\text{(1,1)}\to \text{(1,8)}\to \text{(6,8)}$ and $\text{(1,1)}\to \text{(1,3)}\to \text{(4,3)}\to \text{(4,8)}\to \text{(6,8)}$

Concept Introduction: A differential is said to be exact only if the integral of the function is path independent. (Independent of path means that the system depends only on starting and finishing points and not on the path followed to reach the finishing point.) So,

for a differential, $\text{dz = f(x,y)dx + g(x,y)dy}$ to be exact in then its integral ${\displaystyle \int \text{f(x,y)dx}}\text{ + }{\displaystyle \int \text{g(x,y)dy}}$ is independent of the path.