MATH 4250/5250, Review for Exam 2

The exam will be based on the homeworks (problem sets 8-11), lectures (2/25-4/8), and the textbook (sections 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4).

Here is a list of definitions, theorems, and formulas you should know for the exam. You should know examples of each definition, theorem, or formula. (Examples can be found in the book, in homework, and in class notes.)

Definitions:

Coordinate transformation (see (a) below)
Directional derivative (p. 81)
Shape operator S (p. 84)
Derivative F_* = DF of a map of surfaces (p. 89)
Gauss map (p. 90)
Normal curvature of M in direction u (p. 92)
Gaussian curvature K, mean curvature H (p. 107)
flat surface (p. 110)
minimal surface (p. 110)
principal directions (see (b) below) (principal vectors p. 93)
principal curvatures (p. 93)
umbilic (p. 95)
E, F, G, l, m, n (p. 111)
isometry (see (c) below) (local isometry p. 235)

(a) If x:D -> M and y:E -> M are coordinate charts for the surface M, the coordinate transformation from x to y is the function φ = y^-1x : x^-1y(E) -> y^-1x(D).

(b) The principal directions of a surface M at a point p are the eigenspaces of the shape operator S_p.

(c) If M and N are surfaces, an isometry f:M -> N is a bijective differentiable function such that f^-1 is differentiable and for all curves α(t) on M, the length of α(t) equals the length of f(α(t)).

Theorems:

S_p = 0 for all p implies M is planar (2.2.17)
S is linear, self-adjoint (symmetric) (2.2.10), (2.3.5)
A symmetric matrix has real eigenvalues and orthogonal eigenvectors (2.3.4)
The shape operator is the derivative of the Gauss map (p. 90)
Normal curvature in direction u equals plus or minus curvature of normal section containing u (2.4.3)
Gauss' Theorema Egregium (lectures 4/3 and 4/6)

Formulas:

S(α') . α' = α''. U (2.4.1)

k(u) = cos²θ k₁ + sin²θ k₂ (2.4.11)

l = U . x_uu, m = U . x_uv, n = U . x_vv (3.2.2)

K = (ln - m²)/(EG - F²) (p. 111)

H = (Gl + En - 2Fm)/2(EG - F²) (p. 111)