Bonus (10 pts) Verify that the functions
(a) $f: \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}$,
$f(x, y, z) = (x^2 + y^2 + z^2)^{-1/2}$
(b) $f: \mathbb{R}^3 \to \mathbb{R}$,
$f(x, y, z) = x^2 + xy + 2y^2 - 3z^2 + xyz$
satisfy the Laplace equation
$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0$.
Such functions are called harmonic.
Remark: The differential operator
$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$
is often called the Laplace operator or Laplacian, and also denoted by $\Delta$.
Harmonic functions are important in the study of heat, electricity, waves,
quantum mechanics... In mathematics, harmonic functions are important
in differential geometry, real and complex analysis.
