ΔU

In equations, the function:

 \Delta U  \,

is called the change Δ in internal energy U of a system, in which the system goes from an initial state, at a specific internal energy, Ui, to a final state, at a specific internal energy Uf, quantified as the difference between the internal energy of the system in its final state less the free energy of the system in its initial state:

 \Delta U = U_f - U_i \,

In more detail, internal energy, in its original formulation, as conceived by German physicist Rudolf Clausius in 1850 to 1865, is the sum of the vis viva T and the ergal J or:

 U = T_v + J_e \,

Referring to a change over time, internal energy can be defined as:

 \Delta U = \Delta T_v + \Delta J_e \,

where ΔT is the change in vis viva in going from the system initial state to system final state, and ΔJ is the change in ergal in going from the system initial state to system final state. If the idea of vis viva change is taken literally, the following expression results:

 \Delta T_v = \frac{1}{2} \bigg ( \sum_{i=1}^j m_{f_i} v^2_{f_i} - \sum_{i=1}^k m_{i_i} v^2_{i_i} \Bigg ) \,

which opens up issues as to the mathematics to the idea of vis viva change, which may not be fully mathematically cogent, as formulated here.

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