Question

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.

          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.

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

Added by Michele S.

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