Exact vs Total differential

Question

I'm not sure I understand the difference between the two. Is it correct to claim that a total differential is always exact but not vice versa?

Thanks

Answer 1

The total differential of $f(x,y)$ is the linearization $$ df = f_x dx + f_y dy $$

A differential is referred to as 'exact' if it is total differential of some function $f$.

for example, consider these questions:

• Is the differential $ydx + xdy $ an exact differential? Yes, because $ydx + xdy$ is the total differential of the function $f=xy$
• Is the differential $ y^2 dx + xdy $ an exact differential? NO, because no function $f(x,y)$ exists such that its total differential is $ y^2 dx + xdy $