Blundell Thermal Physics · Problem 11.3

Energy

Physics — Statistical Mechanics · exact-differentials-path-independence

Problem

Consider the differential

\mathrm{d}z = 2xy\,\mathrm{d}x + (x^2 + 2y)\,\mathrm{d}y. \tag{11.39}

Evaluate the integral

\int_{(x_1,y_1)}^{(x_2,y_2)} \mathrm{d}z

along the paths consisting of straight-line segments (i)

(x_1, y_1) \to (x_2, y_1)

and then

(x_2, y_1) \to (x_2, y_2)

. (ii)

(x_1, y_1) \to (x_1, y_2)

and then

(x_1, y_2) \to (x_2, y_2)

. Is

\mathrm{d}z

an exact differential?