The discussion revolves around the concept of path independence of a line integral involving a harmonic function, specifically examining the integral of the form ∫(f_y dx - f_x dy) where del^2(f) = 0. Participants are tasked with demonstrating that this integral is independent of the path taken in a simple region D.
The discussion is ongoing with participants exploring various mathematical approaches and concepts related to the problem. Some guidance has been offered regarding the properties of conservative vector fields and the implications of harmonic functions, but no consensus or resolution has been reached.
Participants are working within the constraints of the problem statement and the definitions of harmonic functions and vector fields, with an emphasis on the conditions for path independence in the context of line integrals.
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.fk378 said: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...
konthelion said: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