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Path Independence

  1. Aug 5, 2008 #1
    1. The problem statement, all variables and given/known data
    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.

    3. 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...
  2. jcsd
  3. Aug 5, 2008 #2
    Remember that for a conservative vector field, [tex]\nabla \times \textbf{F} = 0[/tex].
  4. Aug 5, 2008 #3

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

    I believe you have to use the Second Fundamental Theorem of Calculus
  5. Aug 5, 2008 #4
    Or you can show that [tex]\nabla^2 f = 0 \implies \nabla \times (f_y \textbf{i} - f_x \textbf{j}) = \textbf{0}[/tex].
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