The Fundamental Theorem of Vector Calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.
2.
The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field ( the order of the equations is not casual, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following ).
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