In Finn's Thermal Physics (equation 2.4), the Young modulus $Y$ of a stretched wire with tension $F$ is given to be
$$Y = \frac{L}{A} \left( \frac{\partial F}{\partial L}\right)_T$$
However, usually the Young modulus is defined to be the ratio of stress and strain, specifically
$$Y = \frac{\sigma}{\varepsilon} = \frac{F L_0}{A \Delta L} = \frac{FL_0}{A(L-L_0)}\implies dF = \frac{AY}{L_0} dL$$
this would suggest that the relationship given in Finn's textbook should actually be
$$Y = \frac{L_0}{A} \left( \frac{\partial F}{\partial L}\right)_T$$So I wondered why they have used $L$ instead of $L_0$ in the definition. Is it a mistake? Thanks!