----- > [!proposition] Proposition. ([[the Euler formula]]) > Let $S$ be a [[differentiable Euclidean submanifold|regular surface]] and $p \in S$. The knowledge of the [[principal curvature|principal curvatures]] $k_1, k_{2}$ at $p$ coming from the [[orthonormal basis|orthonormal]] [[eigenbasis]] $e_{1},e_{2}$ allows us to compute easily the [[normal curvature]] along a given direction of $T_{p}(S)$. > Let $v \in T_{p}(S)$ with $\|v\|_{2}=1$; let $\theta$ be the [[angle between vectors|angle]] from $e_{1}$ to $v$ in the [[orientable manifold|orientation]] of $T_{p}(S)$. Then the [[normal curvature]] $k_{n}$ in the direction of $v$ is $k=k_{1} \cos ^{2}\theta + k_{2} \sin ^{2}\theta.$ \ This is known classically as **the Euler formula**; actually, it is just the expression of the [[second fundamental form]] in the [[basis]] $\{ e_{1},e_{2} \}$. > [!proof]- Proof. ([[the Euler formula]]) > First note that $v$ is the [[linear combination]] $\cos \theta \ e_{1} + \sin \theta \ e_{2}$. > > Then it is just computation. $\begin{align} > k= & \text{II}_{p}(v,v) \\ > = & -dN |_{p} (v) \cdot v \\ > = &[ \cos \theta \ dN |_{p} (e_{1}) + \sin \theta \ dN |_{p}(e_{2}) ] \cdot (\cos \theta \ e_{1} + \sin \theta \ e_{2}) \\ > = & \lambda_{1}\cos ^{2}\theta \ + \lambda_{2}\sin ^{2} \theta . > \end{align}$ > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```