# What exactly are the definitions of $\varphi^*dz$ and $\varphi^*d\bar{z}$?

Suppose $\varphi$ is a smooth map on $\mathbb{C}$. For a function $f$, we define a $0$-form $\varphi^*f$ as $\varphi^*f=f\circ\varphi$. Also, $$\varphi^*\,dx=\frac{\partial\varphi_1}{\partial x}\,dx+\frac{\partial\varphi_1}{\partial y}\,dy, \qquad \varphi^*dy=\frac{\partial\varphi_2}{\partial x}\,dx+\frac{\partial\varphi_2}{\partial y}\,dy,$$ where $\varphi_1$ is the $x$ component of $\varphi$ and $\varphi_2$ is the $y$ component of $\varphi$.

I'm curious, are there sensible analogous definitions for $\varphi^*dz$ and $\varphi^*d\bar{z}$ for the complex case?

I know $dz=dx+idy$, so considering it as a $1$ form I thought maybe \begin{align*} \varphi^*dz &= \varphi^*(dz+idy)\\ &= (\varphi^* 1)\varphi^* dx+(\varphi^* i)\varphi^*dy\\ &= \varphi^*dx+i\varphi^* dy \end{align*} but I think it's not right to equate a constant with a constant function. Likewise in the case of $dz=dx-idy$. What is the proper definition for the complex differential?

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In your equations $\phi^*dx=\ldots$, $\phi^*dy=\ldots$ the $dx$, $dy$ on the left side and the $dx$, $dy$ on the right side do not mean the same thing. –  Christian Blatter Apr 23 '12 at 8:23

One can use the Wirtinger derivatives to define $\phi^*dz$. These derivatives are the operators defined by $$\frac{\partial}{\partial z} = \frac 12 \Bigl( \frac{\partial}{\partial x} - i\frac{\partial}{\partial y} \Bigr) \quad \text{and} \quad \frac{\partial}{\partial \overline z} = \frac 12 \Bigl( \frac{\partial}{\partial x} + i\frac{\partial}{\partial y} \Bigr).$$ These are defined as to make $\partial z/\partial z = 1$, $\partial z /\partial \overline z = 0$, and so on. Then we can set $$\phi^* dz = \frac{\partial \phi}{\partial z} dz \quad \text{and} \quad \phi^* dz = \frac{\partial \phi}{\partial \overline z} d\overline z.$$ If you work through the algebra, this should give the same result as your attempt.