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When it comes to a system of linear equations, - it is determined when it has $N$ variables and $N$ linearly independent equations.

If the equations are less, it is underdetermined; if more, it is overdetermined.

What about a system of linear PDEs?

Example: $f = f(x_1,x_2,...,x_N)$

The PDE system contains all first-order derivatives $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, ..., \frac{\partial f}{\partial x_N}$.

The PDE system is linear.

What conditions are to be met by the PDE system for it to have an unique solution?

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