Path Independence of Line Integral: del^2(f)=0

[fk378]

Homework Statement

If f is a harmonic function, that is del^2(f)=0, show that the line integral: (integral)f_y dx - f_x dy is independent of path in any simple region D.

The Attempt at a Solution

I tried to rewrite the given integral as integral of Q dx - P dy, since path independence means vector field F=del f. But I don't know where it's supposed to take me...

 

[foxjwill]

Remember that for a conservative vector field, $$\nabla \times \textbf{F} = 0$$.

 

[konthelion]

Homework Statement

If f is a harmonic function, that is del^2(f)=0, show that the line integral: (integral)f_y dx - f_x dy is independent of path in any simple region D.

The Attempt at a Solution

I tried to rewrite the given integral as integral of Q dx - P dy, since path independence means vector field F=del f. But I don't know where it's supposed to take me...

Let $$f_{x}=M$$, $$f_{y}=N$$ and f is harmonic i.e. $$\bigtriangledown^2f(x,y)=0$$, then if $$f(x,y)= \int_{(x_{0},y_{0}}^{(x,y)}F dr$$ and $$\bigtriangledown f= Mi + Nj$$, then you need to prove that $$\int Ndx - Mdy$$ is path-independent.

I believe you have to use the Second Fundamental Theorem of Calculus

 

[foxjwill]

Let $$f_{x}=M$$, $$f_{y}=N$$ and f is harmonic i.e. $$\bigtriangledown^2f(x,y)=0$$, then if $$f(x,y)= \int_{(x_{0},y_{0}}^{(x,y)}F dr$$ and $$\bigtriangledown f= Mi + Nj$$, then you need to prove that $$\int Ndx - Mdy$$ is path-independent.

I believe you have to use the Second Fundamental Theorem of Calculus

Or you can show that $$\nabla^2 f = 0 \implies \nabla \times (f_y \textbf{i} - f_x \textbf{j}) = \textbf{0}$$.