Notation II: Surfaces

See also this page with basic notation.

Surfaces in $\RR^3$:

$\mathbf{x}:D\longmapsto\RR^3$, with $D\subset\RR^2$

also written as a position vector:

$\xx(u,v) = x(u,v)\,\xhat + y(u,v)\,\yhat + z(u,v)\,\zhat$

Tangent vectors:

$\ev_1 = \frac{\partial\xx}{\partial u}, \ev_2 = \frac{\partial\xx}{\partial v}$

Normal vector:

$\nv = \ev_1\times\ev_2 \ne 0$

Unit normal vector:

$\nn = \frac{\nv}{|\nv|} = \frac{\ev_1\times\ev_2}{|\ev_1\times\ev_2|}$

Shape operator:

$S(\vv) = -\nabla_\vv\nn$

$S(\ev_1) = -\frac{\partial\nn}{\partial u}$, $S(\ev_2) = -\frac{\partial\nn}{\partial v}$

$S^T = S$

Normal curvature:

$k(\vv) = \frac{S(\vv)\cdot\vv}{\vv\cdot\vv}$

Principal curvatures and directions:

$S(\ee_m) = k_m \ee_m$ (no sum)
($\ee_m$ is not necessarily $\frac{\ev_m}{|\ev_m|}$, which is typically only true in orthogonal coordinates.)

Gaussian and mean curvature

$K = k_1 k_2 = \det S$, $H = \frac12 (k_1+k_2) = \frac12 \tr~S$

$k_m = H \pm \sqrt{H^2 - K}$

$S(\vv)\times S(\ww) = K\, \vv\times\ww$
$S(\vv)\times\ww + \vv\times S(\ww) = 2H\, \vv\times\ww$

$S(\ee_m) = \nabla_{\ee_m} \nn = \omega_{3i}(\ee_m) \,\ee_i$ $\qquad$ (If $\ee_m$ is principal, only term in sum is $i=m$.)

$S = \bigl( \omega_{i3}(\ee_j) \bigr)$

Computation:

$E = \frac{\partial\xx}{\partial u}\cdot\frac{\partial\xx}{\partial u}, \quad F = \frac{\partial\xx}{\partial u}\cdot\frac{\partial\xx}{\partial v}, \quad G = \frac{\partial\xx}{\partial v}\cdot\frac{\partial\xx}{\partial v}$
$L % = S(\xx_u)\cdot\frac{\partial\xx}{\partial u} = \nn\cdot\frac{\partial^2\xx}{\partial u^2}, \quad M % = S(\xx_u)\cdot\frac{\partial\xx}{\partial v} = \nn\cdot\frac{\partial^2\xx}{\partial u\,\partial v}, \quad N % = S(\xx_v)\cdot\frac{\partial\xx}{\partial v} = \nn\cdot\frac{\partial^2\xx}{\partial v^2}$
$K = \frac{LN-M^2}{EG-F^2}, \quad H = \frac12\frac{GL-2FM+EN}{EG-F^2}$

Adapted frame:

$\{\ee_1,\ee_2,\nn\}$

Connection on surface:

$d\sigma_1 = \omega_{12}\wedge\sigma_2, \quad d\sigma_2 = \omega_{21}\wedge\sigma_1$
$\omega_{31}\wedge\sigma_1 + \omega_{32}\wedge\sigma_2 = 0$
$d\omega_{12} = \omega_{13}\wedge\omega_{23} = -K \sigma_1\wedge\sigma_2$
$d\omega_{13} = \omega_{12}\wedge\omega_{23}, \quad d\omega_{23} = \omega_{21}\wedge\omega_{13}$