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Consider the Euler Equations $$ \begin{cases} \frac{Du}{Dt} + \left< u, \nabla u \right> = \nabla p, \\\\ \text{div} (u) = 0. \end{cases} $$ I understand how the first equation is derived, and I get what the second equation represents. But, I am having trouble understanding how the second equation plays a role. Specifically, how do I use it when doing finite differences? (I am not just having this problem with Euler's equations and I am fine if you answer the analogous question for Maxwell's equations instead.)

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The second equation plays a role since the unknowns are $p$ and $u$. So you have a system of a vector and a scalar equation and you have three plus one unknowns. Without the second equation, the problem would be under-determined.

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