Writing the Mean Curvature as a Function of Normal Curvature

Question

I am trying to solve a question, and require some assistance. The question reads:

Let $\kappa_n(\theta)$ denote the normal curvature of a surface $M$ in the direction of $\vec{u}=\cos(\theta)\vec{e_1}+\sin(\theta)\vec{e_2}$, where $\vec{e_1}$ and $\vec{e_2}$ are principal tangent vectors in the tangent plane $T_{p}M$. Show that the mean curvature $H(p)$ of $M$ at $p$ is given by: $$H(p)=\frac{1}{2\pi}\int_0^{2\pi}\kappa_n(\theta)d\theta$$

Now, I know that $\kappa^2=\kappa^2_n+\kappa^2_g$. Am I supposed to compute $H$ and $\kappa_n$ directly using $u$? Or is there an identity that I am not aware of?

Answer 1

HINT: Use Euler's formula for $\kappa_n$ as a function of $\theta$ (and the definition of mean curvature, of course).