Thermal Expansion

$$\begin{array}{rl}& \\ & \Delta L=L_0{\alpha }_L\Delta T\\ & \Delta V=V_0{\alpha }_L\Delta T\\ & \\ & {\alpha }_V=\frac{1}{V}\frac{dV}{dT}={\alpha }_x+{\alpha }_y+{\alpha }_z\\ & {\alpha }_L=\frac{1}{L}\frac{dL}{dT}\end{array}$$