# For $u$ harmonic, $\ast du = \frac{\partial u}{\partial n} |dz|$?

I don't understand a statement of Ahlfors's Complex Analysis on page 163 in Harmonic Functions:

If $u$ is a harmonic function and $\gamma$ is a regular curve with equation $z=z(t)$ and $\frac{\partial}{\partial n}$ is the normal that points to the right of the tangent, then we obtain $\ast du = \frac{\partial u}{\partial n} |dz|$. Here, $\ast du$ means $-\frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x}dy$.

I don't understand this statement at all because the left hand side and right hand side don't seem to be the same object at all. $\ast du$ is a one-form, but $\frac{\partial u}{\partial n} |dz|$ seems to be a number $\frac{\partial u}{\partial n}$ times a metric $|dz|$. Right-hand side doesn't look like a 1-form to me. Can anyone explain what Ahlfors means?

## 1 Answer

Of course they are different things. $\ast du=-u_ydx+u_xdy$ is defined on $\Omega$ whenever $u_x, u_y$ exist. While on the other hand $\frac{\partial u}{\partial n} |dz|$ is not defined if a regular curve $\gamma$ is not given. When $\gamma$ is not given, "normal derivative with respect to $\gamma$" has no meanings.

But once $\gamma$ is given, $\frac{\partial u}{\partial n} |dz|$ can be defined on $\gamma$ and

\begin{align} &\frac{\partial u}{\partial n}=u_x\cos \beta +u_y\sin\beta =u_x\sin\alpha -u_y\cos \alpha ,\\ &\frac{\partial u}{\partial n} |dz|=u_x|dz|\sin \alpha -u_y|dz|\cos \alpha =u_xdy-u_ydx, \end{align} since $|dz|\cos \alpha =dx,$ $|dz|\sin \alpha =dy$. ($\alpha , \beta$ are the same as in Ahlfors p.163.)

Therefore $$\ast du = \frac{\partial u}{\partial n} |dz|$$ holds on $\gamma$ and $$\int_\gamma \ast du = \int_\gamma \frac{\partial u}{\partial n} |dz|$$ holds for every regular curve $\gamma$.