# smooth manifolds – do Carmo Differential forms and application Chapter 2 Exercise 6

Line integrals are quite useful in the study of complex functions $$f:Bbb Cto Bbb C$$. Here the complex plane $$Bbb C$$ is identified with $$Bbb R^2$$ by setting $$z=x+iy$$, $$zinBbb C$$, $$(x,y)inBbb R^2$$. It is convenient to introduce the complex differential form $$dz = dx+idy$$ and to write
$$f(z) =u(x,y)+iv(x,y) = u+iv.$$
Then the complex form
$$f(z)dz = (u+iv)(dx+idy) = (udx-vdy)+i(udy+vdx)$$
has $$udx-vdy$$ as its real part and $$udy+vdx$$ as its imaginary part. Define
$$int_c f(z) dz = int_c(udx-vdy)+iint_c(udy+vdx)$$
Assume that $$u,vin C^1$$. Recall that $$f$$ is holomorphic if and only if $$u_x = v_y, u_y = -v_x$$

Question. If $$f$$ is holomorphic, the function $$f'(z)$$ (the derivative of $$f$$ in $$z$$) given by the equation $$df:=du+idv = f'(z)dz$$ is well defined and $$f'(z) = u_x-iu_y$$.

In the question I what should I check for well definedness?