Kelvin Transform

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Problem 3308 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

Prove that if the function $u=u(x,y)$ satisfies the Laplace equation (see problem 3307: $\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}=0$ ), then the function

v=u\left(\dfrac {x}{x^{2}+y^{2}}, \dfrac {y}{x^{2}+y^{2}}\right)

also satisfies this equation.

Problem 3312 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.2 "Partial Derivatives. Differentials of Functions of Several Variables"

Prove that if the function $u=u(x,y,z)$ satisfies the Laplace equation (see problem 3311: $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}=0$ ), then the function

v=\dfrac {1}{r}u\left(\dfrac {k^{2}x}{r^{2}}, \dfrac {k^{2}y}{r^{2}}, \dfrac {k^{2}z}{r^{2}}\right)

(where $k$ is a constant and $r=\sqrt{x^{2}+y^{2}+z^{2}}$ ) also satisfies this equation.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.