Discussion

Any function that gives an extremum of the functional

\[ \iint_{D} \{z_{x}^{2} + z_{y}^{2} + 2zf(x, y)\}\,dx\,dy \]

must satisfy :

Answer: D

The extremal must satisfy Poisson's equation.

For \(F(z, z_{x}, z_{y}) = z_{x}^{2} + z_{y}^{2} + 2zf(x, y)\), the Euler-Ostrogradsky equation is

\[ \begin{aligned} F_{z} - \frac{\partial}{\partial x}F_{z_{x}} &- \frac{\partial}{\partial y}F_{z_{y}} \\ &= 0 \end{aligned} \]

which gives \(2f - 2z_{xx} - 2z_{yy} = 0\), that is

\[ \nabla^{2}z = f(x, y) \]

That is Poisson's equation. It reduces to Laplace's equation only if \(f \equiv 0\), and there is no time variable for a heat or wave equation.

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