# Proving well defined (complex variables)

1. Oct 20, 2007

### Milky

1. The problem statement, all variables and given/known data

Let D be a connected domain in R^2 and let u(x,y) be a continuous vector field defined on D. Suppose u has zero circulation and zero flux for any simple closed contour on D.

$$u(x,y) = (u_1(x,y),u_2(x,y))$$

$$\Gamma = \int_{c}(u\circ\gamma)tds = 0$$

$$F=\int_{c}(u\circ\gamma)nds$$ = 0[/tex]

$$\phi(x,y)=\int_{c}(u\circ\gamma)tds$$

$$\psi(x,y)=\int_{c}(u\circ\gamma)nds$$

Prove that $$\phi, \psi$$ are well defined.

3. The attempt at a solution
For $$\phi$$:

I think to prove its well defined means to prove that if $$(x,y)=(x_0,y_0), then \phi(x,y)=\phi(x_0,y_0)$$

By Cauchy Formula, the integral of one path is equal to the integral of another:

$$\int_{c}(u\circ\gamma)tds = 0 = \int_{c_2}(u\circ\gamma)tds - \int_{c_1}(u\circ\gamma)tds$$

Then, $$\int_{c_2}(u\circ\gamma)tds$$ = $$\int_{c_1}(u\circ\gamma)tds$$

So, when $$(x,y)=(x_0,y_0)$$ the integrals are equal as well.
Is this how to prove it is well-defined?

Last edited: Oct 20, 2007