Math 109b - Final • Deposit your finished exam into the homework box by 5pm on March 15, 2017 (Wednesday). • You have three hours to finish all five problems. • Write your solutions in a blue book. • Write down the time and date you started the exam and the time and date you finished the exam on the cover of your blue book. • You may refer ONLY to your class notes and homework. • Do not look at the problems or discuss the problems with anybody before taking the exam. Good luck!

1

1. (22 points) Let S be the genus 2 surface. (a) (8 points) Compute the Euler characteristic of S. (b) (6 points) Show that every smooth vector field on S must have a singular point. (c) (8 points) Show that for every smooth immersion φ : S → R3 , there is a point p ∈ S where the principal curvatures are of opposite signs. p 2. (16 points) Let T2 := {(x, y, z) ∈ R3 : ( x2 + y 2 − 2)2 + z 2 = 1}, i.e. T2 is the surface obtained by rotation the circle {(x, y, z) : (x2 − 2)2 + z 2 = 1, y = 0} (in the y = 0 plane in R3 ) about the z-axis. Also, let φ = id : T2 → R3 . (a) (8 points) Let γ : [0, 4π] → T2 be the smooth curve given by γ : t 7→ (2 cos 2t , 2 sin 2t , 1). Compute the absolute value of the geodesic curvature of γ at every t ∈ [0, 4π]. (b) (8 points) Find a geodesic of the immersion φ, and show that it is a geodesic. 3. (16 points) Let φ : R2 → R3 be a smooth immersion and let h·, ·i be the induced Riemannian metric on R2 . Suppose that for all (x, y) ∈ R2 ,   ∂ ∂ , = y 2 + 1, ∂x (x,y) ∂x (x,y) (x,y)  ∂ ∂ , = 0, ∂x (x,y) ∂y (x,y) (x,y)   ∂ ex ∂ , = . ∂y (x,y) ∂y (x,y) (x,y) y 2 + 1 

(a) (8 points) Show that K(x, y), the Gaussian curvature of φ at the point (x, y), is negative for all (x, y) ∈ R2 . (b) (8 points) Let p1 and p2 be a pair of distinct points in R2 . Show that all geodesics γ : I → R2 of φ between p1 and p2 have the same image. 4. (30 points) In this problem, S is connected, orientable, and has a smooth immersion S → R3 . In particular, S is equipped with a Riemannian metric and we can define a covariant derivative on S. Are the following statements true or false? If it is true, give a proof. If it is false, give a counter example. (a) (10 points) If p is an isolated singular point of a smooth vector field on S, then its index cannot be zero. (b) (10 points) Let φ1 : S1 → R3 and φ2 : S2 → R3 be two immersions of smooth surfaces. If there is an isometry f : S1 → S2 (Si is equipped with the metric induced by φi ), then for any p ∈ S1 , the mean curvature at p is equal to the mean curvature at f (p). (c) (10 points) Let γ : I → S be a smooth curve and let σ, τ : I → T S given by σ : t 7→ (γ(t), v(t)) and τ : t 7→ (γ(t), w(t)) be unit vector fields along γ so that d dt hv(t), w(t)i = 0. Then σ is parallel along γ if and only if τ is parallel along γ. 5. (16 points) Let γ : [0, 2π] → S2 be the smooth curve given by γ : t 7→ (cos(t), sin(t), 0). 2

(a) (8 points) Find a smooth vector field σ : S2 → T S2 so that • σ has a unique singular point q, • q does not lie in γ(I). (b) (8 points) Let X be a vector field on S2 . Show that there is some t ∈ [0, 2π] so that X(γ(t)) = γ 0 (t).

3

Math 109b - Final

Mar 15, 2017 - (a) (10 points) If p is an isolated singular point of a smooth vector field on S, then its index cannot be zero. (b) (10 points) Let φ1 : S1 → R3 and ...

90KB Sizes 0 Downloads 70 Views

Recommend Documents

Finite math final review.pdf
1) Mr. Johnson has 8 employees at his landscaping business: Abel, Brian, .... a) What is the probability that a randomly chosen apple has spots? .... We package small boxes of cherries, which we sell to grocery stores, advertised as 12 ounces.

Math 114-Review Final Exam
The equation that models its height, h feet, off the ground t seconds after it was fired is h = - 16t2 + 40t. (a) How high is the rocket 1.5 seconds after it was fired?

Final final final final draft Standard ECMA-262 5th ... - Ecma International
requests, clients, and files; and mechanisms to lock and share data. By using ...... a.i has been performed this loop does not start at the beginning of B) a.

Sem. 2 Final Exam Skills Review Part 1--math analysis.pdf ...
Sem. 2 Final Exam Skills Review Part 1--math analysis.pdf. Sem. 2 Final Exam Skills Review Part 1--math analysis.pdf. Open. Extract. Open with. Sign In.

Final final final final draft Standard ECMA-262 5th ... - Ecma International
... on several originating technologies, the most well known being JavaScript ..... For the purposes of this document, the following terms and definitions apply.

Hora Santa final- final .pdf
Page 1 of 4. Pastoral Vocacional - Provincia Mercedaria de Chile. Hora Santa Vocacional. Los mercedarios nos. consagramos a Dios,. fuente de toda Santidad.

Final Amherst Private School Survey (final).pdf
Choice, Charter, and Private School Family Survey. Page 4 of 33. Final Amherst ... ey (final).pdf. Final Amherst ... ey (final).pdf. Open. Extract. Open with. Sign In.

Final Judgment.pdf
IT IS FURTHER ORDERED, ADJUDGED, and DECREED that Texas. Administrative Code, Title 16, Sections 45.77, 45.79(f), 45.82(f), 45.90, and 45.1 10(c)(3).

final-nirnaya.pdf
kl/dflh{t x'g' clt cfjZos b]lvG5 . lgodgsf/L e"ldsf lgjf{x ug]{ ;DaGwdf sxL st} cK7\of/f]. b]lv+Pdf Gofok"0f{ tj/af6 tTsfn} ;d:ofsf] ;dfwfg ug]{ bfloTj klg ;/sf/sf] g} xf] .

Final Assignment - GitHub
October 9, 2016. The final assignment is flexible. The primary ... Once you select a dataset create a Jupyter notebook or Python script that performs the following:.