$${\left( {\frac{{\partial {\text{E}}}}{{\partial {\text{T}}}}} \right)_{\text{V}}}$$  is the mathematical expression for

By the definition of $${C_V}$$ we can write $${C_V} = \mathop {\lim }\limits_{\delta T \to 0} {\left( {\frac{{\delta Q}}{{\delta T}}} \right)_V}.$$
Since, under constant volume process the expansion work is zero, and by first law of thermodynamics
$$\eqalign{ & \delta Q = dU \cr & \Rightarrow {\left( {\frac{{\partial U}}{{\partial T}}} \right)_V} = {C_V}. \cr} $$