:frown: Normal curvature integral proof

In summary, the task is to prove that the mean curvature H at a point p on a surface S can be calculated using the formula 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 making an angle theta with a fixed direction. The normal curvature, k_n, is defined as k \cdot cos(\theta), where k is the maximum or minimum normal curvature and \theta is the angle between the eigenvectors e1 and e2 of the differential map dN_{p}. The formula for mean curvature can also be written as H = \frac{k_1
  • #1
Mathman23
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Homework Statement



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

[tex]H = \frac{1}{\pi} \cdot \int_{0}^{\pi} k_n{\theta} d \theta[/tex]

where [tex]k_n{\theta}[/tex] 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 [tex]H = \frac{k_1 + k_2}{2}[/tex]
where k1 and K2 are the maximum and minimum normal curvature.

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

The Attempt at a Solution



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

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
 
Last edited:
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  • #2
undeletable
 

1. What is the normal curvature integral in geometry?

The normal curvature integral is a mathematical formula used to calculate the curvature of a surface at a specific point. It takes into account the Gaussian curvature and the mean curvature, which are both measures of how much a surface bends at a given point.

2. Why is the normal curvature integral important?

The normal curvature integral is important because it allows us to understand the curvature of a surface at a specific point, which is crucial in many fields such as physics and engineering. It also helps us to classify and analyze different types of surfaces.

3. How is the normal curvature integral calculated?

The normal curvature integral is calculated by taking the dot product of the surface's normal vector and the Hessian matrix, which contains second-order partial derivatives of the surface's equation. This value is then divided by the determinant of the metric tensor, which is a measure of the surface's curvature.

4. What is the proof for the normal curvature integral?

The proof for the normal curvature integral involves using differential geometry and multivariable calculus concepts such as the first and second fundamental forms, the normal vector, and the Hessian matrix. It is a complex mathematical proof that requires a deep understanding of these concepts.

5. How is the normal curvature integral used in real-world applications?

The normal curvature integral is used in various real-world applications, such as determining the stress and strain on surfaces in engineering and understanding the curvature of Earth's surface in geodesy. It also plays a crucial role in computer graphics and animation, where it is used to create realistic 3D models of surfaces.

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