I am just curious what would be the case in which a vector field ($\vec f :\Bbb R^2 \rightarrow \Bbb R^2$) is well-defined and continuously differentiable on a region R enclosed by a simple closed curve (So R is simply connected), yet $\frac {df_2}{dx} \neq \frac {df_1}{dy}$.

I guess my confusion comes from my complex analysis class in which all functions that are analytic in R (as defined above) will satisfy Cauchy's integral theorem due to Cauchy-Riemann equation.

So, in other words, I would like to see how Cauchy-Riemann equation fails in real analysis.

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    $\begingroup$ Note that being differentiable as a complex function is a much stronger property than being differentiable as a function on $\mathbb{R}^2$. The reason is that if $f$ is differentiable as a complex function, the corresponding jacobi matrix is a scaled rotation, i.e has the form $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$. $\endgroup$ – fgp Apr 8 '14 at 21:11

Try, for example, $$f_1(x,y) =y,$$ $$f_2(x,y) =2x.$$

  • $\begingroup$ simple and permanently unforgettable, thanks a lot! $\endgroup$ – user2386986 Apr 8 '14 at 21:19
  • $\begingroup$ @user2386986 you are welcome! $\endgroup$ – TZakrevskiy Apr 9 '14 at 7:06

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