   Chapter 16, Problem 23RE

Chapter
Section
Textbook Problem

If f is a harmonic function, that is, ∇2f = 0, show that the line integral ∫ fy dx − fx dy is independent of path in any simple region D.

To determine

To show: The line integral fydxfxdy is independent of path in any simple region D.

Explanation

Given data:

2f=0 (1)

Formula used:

Write the expression Green’s Theorem.

CPdx+Qdy=D(QxPy)dA (2)

Consider the expression as follows.

F(x,y)=fyifxj (3)

Consider the expression as follows.

F(x,y)=Pi+Qj (4)

Compare equations (3) and (4).

P=fyQ=fx

Substitute fy for P and fx for Q in equation (2),

Cfydxfxdy=D((fx)x(fy)y)dA=D(</

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