:frown: Normal curvature integral proof

[Mathman23]

Homework Statement

I need to show that the mean curvature H at p \in S given by

H = \frac{1}{\pi} \cdot \int_{0}^{\pi} k_n{\theta} d \theta

where k_n{\theta} is the mean curvature at p along a direction makin an angle theta with a fixed direction.

Homework Equations

I know that the formel definition of the mean curvature is H = \frac{k_1 + k_2}{2}
where k1 and K2 are the maximum and minimum normal curvature.

I know that the normal curvature is defined as k_n = k \cdot cos(\theta). where \theta is defined as the angle between the eigenvectors e1 and e2 of dN_{p}

The Attempt at a Solution

do I then claim that <dN_{p}(\theta), \theta)> = k_n \cdot \theta ??

Could somebody please help me along here? with a hint or something? or this there is some theory that I have missed here??

Best regards
Fred

 

[Mathman23]

undeletable