## frank0520 one year ago 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.

1. frank0520

Here is a picture of the question.

2. ganeshie8

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

3. ganeshie8

$\text{Perp(f)} = \langle -f_y,~f_x\rangle$ find the curl

4. anonymous

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

5. anonymous

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. $$f_{xx}+f_{yy}=0$$)

6. anonymous

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