Exercise · ZW-HARMONIC-FUNCTION-010 · ★★★ Advanced
If \(u\) and \(v\) are harmonic on an open set, is \(uv\) always harmonic?
Choose one option.
Show answer
- C. No: with \(u=v=x\), \(uv=x^2\) has \(\Delta(x^2)=2\)
Explanation
Linearity only gives that \(u+v\) is harmonic. The product also brings in \(2(u_xv_x+u_yv_y)\), which does not vanish in general; with \(u=v=x\) it equals \(2\).
Review the concept: Harmonic function
A real function \(u\) of two variables, with continuous second partial derivatives on an open set \(U\) of the plane, is harmonic if its Laplacian \(\Delta u\) is zero throughout \(U\).
\[\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\]
More practice problems: harmonic function
- ZW-HARMONIC-FUNCTION-001 Which of these functions is harmonic on the whole plane?
- ZW-HARMONIC-FUNCTION-002 Compute Delta u for u=x^3+y^3.
- ZW-HARMONIC-FUNCTION-003 A harmonic conjugate of u=x^2-y^2 is
- ZW-HARMONIC-FUNCTION-004 A harmonic conjugate of u=e^xcos y is
- ZW-HARMONIC-FUNCTION-005 Which of these functions can be the real part of a holomorphic function on all of ℂ?
- ZW-HARMONIC-FUNCTION-006 Find a so that u=x^2+ay^2 is harmonic.
- ZW-HARMONIC-FUNCTION-007 Find a>0 so that u=e^axsin y is harmonic.
- ZW-HARMONIC-FUNCTION-008 Find a harmonic conjugate of u=x^3-3xy^2.