Geometry | Part IB, 2005

Given a parametrized smooth embedded surface σ:VUR3\sigma: V \rightarrow U \subset \mathbf{R}^{3}, where VV is an open subset of R2\mathbf{R}^{2} with coordinates (u,v)(u, v), and a point PUP \in U, define what is meant by the tangent space at PP, the unit normal N\mathbf{N} at PP, and the first fundamental form

Edu2+2Fdudv+Gdv2.E d u^{2}+2 F d u d v+G d v^{2} .

[You need not show that your definitions are independent of the parametrization.]

The second fundamental form is defined to be

Ldu2+2Mdudv+Ndv2,L d u^{2}+2 M d u d v+N d v^{2},

where L=σuuN,M=σuvNL=\sigma_{u u} \cdot \mathbf{N}, M=\sigma_{u v} \cdot \mathbf{N} and N=σvvNN=\sigma_{v v} \cdot \mathbf{N}. Prove that the partial derivatives of N\mathbf{N} (considered as a vector-valued function of u,vu, v ) are of the form Nu=aσu+bσv\mathbf{N}_{u}=a \sigma_{u}+b \sigma_{v}, Nv=cσu+dσv\mathbf{N}_{v}=c \sigma_{u}+d \sigma_{v}, where

(LMMN)=(abcd)(EFFG)-\left(\begin{array}{cc} L & M \\ M & N \end{array}\right)=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{ll} E & F \\ F & G \end{array}\right)

Explain briefly the significance of the determinant adbca d-b c.

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