Question

Suppose f(x,y) is harmonic (f has continuous second partials and fxx +fyy =0) and define the vector field
Perp(f)=<-fy,fx>
a) Show that Perp(f) is a conservative vector field that is orthogonal to Grad(f).
Hint: To show that Perp(f) is conservative use one of the equivalent properties.
b) Since Perp(f) is conservative it has a potential function g(x,y) called the harmonic conjugate of f(x,y). Show that g(x,y) is in fact harmonic.

Answer 1

Here is a picture of the question.

Answer 2

For part a, use the given hint :
If the vector field is conservative, then the curl is 0

Answer 3

\[\text{Perp(f)} = \langle -f_y,~f_x\rangle\]

find the curl

Answer 4

if \(\nabla g=\langle -f_y,f_x\rangle\) then it follows \(g_x=-f_y,g_y=f_x\) so it follows $$g_{xx}=\frac{\partial}{\partial x}(-f_y)=-f_{xy}\\g_{yy}=\frac{\partial}{\partial y}(f_x)=f_{yx}$$so it follows by symmetry of mixed partials \(f_{xy}=f_{yx}\) that \(g_{xx}+g_{yy}=-f_{xy}+f_{yx}=0\), i.e. \(g\) is indeed harmonic

Answer 5

to prove \(\langle -f_y,f_x\rangle=\langle -f_y,f_x,0\rangle\) as a vector field on \(\mathbb{R}^3\) is conservative, we can show \(\nabla\times f=\langle0,0,\frac{\partial}{\partial x}(f_x)-\frac{\partial}{\partial y}(-f_y)\rangle=\langle0,0,f_{xx}+f_{yy}\rangle=0\) since we're told \(f\) is harmonic (i.e. \(f_{xx}+f_{yy}=0\))

Answer 6

to show it's orthogonal to \(\nabla f=\langle f_x,f_y\rangle\) you can compute their dot product: $$\langle -f_y,f_x\rangle\cdot\langle f_x,f_y\rangle=-f_yf_x+f_xf_y=0$$