MATH 434 Fall 2016 Homework 1, due on Wednesday August 31

Size: px
Start display at page:

Download "MATH 434 Fall 2016 Homework 1, due on Wednesday August 31"

Transcription

1 Homework 1, due on Wednesday August 31 Problem 1. Let z = 2 i and z = 3 + 4i. Write the product zz and the quotient z z in the form a + ib, with a, b R. Problem 2. Let z C be a complex number, and let z be its conjugate. Show that z is a real number if and only if z = z. Namely: a. First show that, if z is a real number, then z = z. b. Then show that, if z = z, then z is a real number. Problem 3. Find r and θ so that i 1 = re iθ. Hint: First plot i 1 in the complex plane, and use polar coordinates. Problem 4. Let ϕ: C C be the rotation of angle θ around the point z 0 C. Express ϕ(z) in terms of z, z 0 and e iθ. (Remember that we considered the case z 0 = 0 in class.) Problem 5. The map ψ : C C defined by ψ(z) = z is a relatively simple transformation of the plane. What is it? (Namely describe it with words, such as the rotation of angle π 7 around the point 2 i ; of course, this is not the answer.) 1

2 Homework 2, due on Friday September 9 Problem 1. Let (X, d) be a metric space. a. Show that d(p, Q) d(p, Q ) d(q, Q ) for every P, Q, Q X. b. Conclude that d(p, Q) d(p, Q ) d(q, Q ) for every P, Q, Q X. Problem 2. Let X be the plane R 2, and let d 1, d 2, d 3 : X X R be defined by d 1 ( (x, y), (x, y ) ) = (x x ) 2 + (y y ) 2 d 2 ( (x, y), (x, y ) ) = x x + y y d 3 ( (x, y), (x, y ) ) = max{ x x, y y }. In particular, d 1 is the usual euclidean distance d euc, and we proved in class that (X, d 1 ) is a metric space. a. Show that (X, d 2 ) is a metric space. b. Show that (X, d 3 ) is a metric space. Problem 3. In a metric space (X, d), the ball of radius r centered at the point P is the set B d (P, r) = {Q X; d(p, Q) < r} consisting of all points Q in X such that d(p, Q) < r. For the metric spaces (X, d 1 ), (X, d 2 ) and (X, d 3 ) of Problem 2 and for the point P 0 = (0, 0) in X = R 2, draw the balls B d1 (P 0, 1), B d2 (P 0, 1) and B d3 (P 0, 1).

3 Homework 3, due on Wednesday September 14 Problem 1. In the hyperbolic plane H 2, consider the two points P = i and Q = 4 + i. For u > 0, let P u = ui, let Q u = 4 + ui, and let γ u be the curve going from P to Q that is made up of the vertical line segment [P, P u ], followed by the horizontal line segment [P u, Q u ], and finally followed by the vertical segment [Q u, Q]. a. Draw a picture of γ u. b. Compute the hyperbolic length l hyp (γ u ). c. For which value of u is l hyp (γ u ) minimum? (Hint: Remember calculus?) d. Use Part c to show that d hyp (P, Q) 2 ln Problem 2. Let ϕ: H 2 H 2 be the map defined by the property that ϕ(x, y) = ( x, y). (Namely, ϕ is the euclidean reflection across the y axis.) a. Show that, if γ is a curve in H 2 and if γ 1 is the image of γ under ϕ, then l hyp (γ 1 ) = l hyp (γ). b. Use Part a to show that ϕ is an isometry from (H 2, d hyp ) to itself.

4 Homework 4, due on Wednesday September 21 Problem 1. Given four numbers a, b, c, d C with ad bc = 1 consider the map ϕ(z) = az + b cz + d defined for any complex number z different from d c. a. Given a similar map b. If ϕ (z) = a z + b c z + d with a, b, c, d C and a d b c = 1, compute the composition ϕ ϕ and show that there exists a, b, c, d C with a d b c = 1 such that ϕ ϕ (z) = a z + b c z + d for every z where it is defined. ψ(z) = compute ϕ ( ψ(z) ) and ψ ( ϕ(z) ). dz b cz + a, Remark. (No credit) If you remember from linear algebra how to multiply matrices, you may notice that (a ) ( ) ( ) b a b a b c d = c d c d. This is not a coincidence. (Do not write anything. This is just intended to whet your appetite for more math.) Problem 2. Let ϕ(z) = az + b with a, b, c, d R and ad bc = 1 cz + d as in Problem 1, and suppose in addition that a 0. Note that a, b, c, d are now real numbers. Set ϕ 1 (z) = z + b a ϕ 2 (z) = 1 z ϕ 3 (z) = 1 a 2 z ϕ 4(z) = z + c a. a. Which ones of ϕ 1, ϕ 2, ϕ 3, ϕ 4 are horizontal translations, homotheties or inversions? b. Show that ϕ = ϕ 2 ϕ 4 ϕ 3 ϕ 2 ϕ 1. c. Show that ϕ defines an isometry of the hyperbolic plane (H 2, d hyp ). (Hint: Part b.)

5 Homework 5, due on Wednesday September 29 Problem 1. Inspired by what we did in class for isometries of the hyperbolic plane (H 2, d hyp ), the goal of this problem is to describe all isometries of the euclidean plane (R 2, d euc ). More precisely, we will rigorously prove that all isometries of (R 2, d euc ) are the ones we saw in class a few weeks ago, and the proof will be cut into several steps. In particular, each question usually relies on the previous ones. a. Consider the two points P 1 = (0, 0) and P 2 = (1, 0). Show that, for any two positive numbers d 1 and d 2, there exists exactly zero, one or two points P = (x, y) such that d euc (P, P 1 ) = d 1 and d euc (P, P 2 ) = d 2. When they are two such points, show that they are related to each other by reflection across the x axis. (Hint: Express d euc (P, P 1 ) and d euc (P, P 2 ) in terms of x and y, and solve.) b. Consider in addition the point P 3 = (0, 1). Show that if the two points P = (x, y) and P = (x, y ) are such that d euc (P, P 1 ) = d euc (P, P 1 ), d euc (P, P 2 ) = d euc (P, P 2 ) and d euc (P, P 3 ) = d euc (P, P 3 ), then necessarily P = P. c. Let ϕ: R 2 R 2 be an isometry of (R 2, d euc ) such that ϕ(p 1 ) = P 1, ϕ(p 2 ) = P 2 and ϕ(p 3 ) = P 3. Show that ϕ(p ) = P for every P R 2. d. Let ϕ: R 2 R 2 be an isometry of (R 2, d euc ) such that ϕ(p 1 ) = P 1 and ϕ(p 2 ) = P 2. Show that ϕ is, either the identity map defined by ϕ(x, y) = (x, y), or the reflection ϕ(x, y) = (x, y) across the x axis. e. Let ϕ: R 2 R 2 be an isometry of (R 2, d euc ). Show that there exists a translation ψ 1 that sends ϕ(p 1 ) to P 1. Show that there exists a rotation ψ 2 around the point P 1 = (0, 0) such that ψ 2 (ψ 1 ( ϕ(p2 ) )) = P 2. (You may need to use the fact that ψ 1 and ϕ are isometries.) Show that, for the composition ψ = ψ 2 ψ 1, there exists z 1 C and an angle θ 1 R such that, in complex coordinates, ψ(z) = e iθ1 z + z 1. f. For ϕ and ψ as in Part e, show that the composition ψ ϕ is an isometry of (R 2, d euc ) that sends P 1 to P 1, and sends P 2 to P 2. g. Combine Parts d, e and f (and a short computation) to show that, for every isometry ϕ of the euclidean plane (R 2, d euc ), there exists z 0 C and θ R such that, either ϕ(z) = e iθ z + z 0 for every z C, or ϕ(z) = e 2iθ z + z 0 for every z C. Problem 2. Problem 1 is long enough. There is no Problem 2.

6 Math 434 Practice Midterm The actual exam will have one fewer problem. Do not turn this in. Problem 1. Consider the two points P = ( 2, 2) and Q = (2, 2) in the hyperbolic plane (H 2, d hyp ). a. Compute the hyperbolic length l hyp ( [P, Q] ) of the line segment [P, Q]. b. What is the shortest curve going from P to Q (where shortest means shortest for the hyperbolic arc length l hyp )? c. Give a parametrization of this shortest curve from P to Q. d. Compute the hyperbolic distance d hyp (P, Q). Problem 2. On a set X, define for every two points P, Q X. d(p, Q) = { 0 if P = Q 1 if P Q Show that (X, d) is a metric space. (Remember that there are four conditions to check.) Problem 3. Let ϕ: X X be an isometry of the metric space (X, d), such that ϕ(p 0 ) = P 0 for some point P 0 X. Show that ϕ sends each P X to a point ϕ(p ) that is at the same distance from P 0 as P, namely such that d ( ϕ(p ), P 0 ) = d(p, P0 ). Problem 4. a. Show that (cos θ + i sin θ) 5 = cos 5θ + i sin 5θ for every θ R. Hint: e iθ. b. Use Part a to show that for every θ R. cos 5θ = cos 5 θ 10 cos 3 θ sin 2 θ + 5 cos θ sin 4 θ Problem 5. Let ϕ: H 2 H 2 be the isometry of (H 2, d hyp ) defined by ϕ(z) = az + b with a, b, c, d R and ad bc = 1. cz + d Suppose in addition that a + d > 2 and c 0. a. Show that there exists exactly two points x R such that ϕ(x) = x. Hint: quadratic formula. b. Use Part a to show that there is a unique complete geodesic g in H 2 such that ϕ(g) = g.

7 Math 434 Actual midterm The percentages denote the percentage of points assigned to each problem/subproblem. Problem 1. (Total: 20%) a. (10%) Give the x and y coordinates of the point corresponding to the complex number z = 2e i π 4. b. (10%) Find r and θ such that 1 + i 3 = re iθ. Problem 2. (Total: 30%) Consider the points P = (1, 3) and Q = (0, 2) in the hyperbolic plane (H 2, d hyp ). a. (6%) What are the polar coordinates of P and Q? b. (8%) What is the shortest curve going from P to Q (where shortest means shortest for the hyperbolic length l hyp )? c. (8%) Give a parametrization of this shortest curve from P to Q. d. (8%) Express the hyperbolic distance d hyp (P, Q) as an integral of explicit functions, but do not try to compute this integral. (Namely, leave your answer as something like d hyp (P, Q) = 13 π 7 ln t3 + cos 5t sin 3 t + 5 dt.) Problem 3. (Total: 25%) Let f : R R be any positive continuous function defined on the real line R (namely, f(x) > 0 for every x R). Define a function d: R R R of two variables by sup{f(z); x z y} if x < y d(x, y) = sup{f(z); y z x} if y < x 0 if x = y. Namely, d(x, y) is the supremum of the values taken by f between x and y. Show that (R, d) is a metric space. (It may be useful to remember the Extreme Value Theorem from calculus, which says that the function f achieves its maximum over each closed interval [a, b]; namely, for every closed interval [a, b], there exists c [a, b] such that f(c) = sup{f(x); x [a, b]}.) Problem 4. (Total: 25%) Let ψ: H 2 H 2 be the (antilinear fractional) isometry of (H 2, d hyp ) defined by ψ(z) = c z + d with a, b, c, d R and ad bc = 1. a z + b Suppose in addition that b + c = 0 and a 0. Show that the set of points z H 2 such that ψ(z) = z is a complete geodesic, namely a semi-circle centered on the x axis. (You may find it convenient to switch to cartesian coordinates after the preliminary steps of the computation.)

8 Homework 6, due on Wednesday October 19 Recall from several weeks ago that, in a metric space (X, d), the ball of radius r centered at P X is B d (P, r) = {Q X; d(p, Q) < r}. The three problems are devoted to these balls, in various spaces. Note that the assignment continues on the next page. Problem 1. We first consider the disk model (B 2, d B 2). Let 0 be the center of the disk B 2. ) a. For a point P B 2, express the B 2 length l B 2( [0, P ] of the line segment [0, P ] in terms of the euclidean distance D = d euc (0, P ). In the, I know, very unlikely event that you forgot about partial fractions I remind you that dx 1 x 2 = 1 2 dx 1 + x dx 1 x = 1 2 ln 1 + x 1 x + C. b. For 0 and P as in Part a, what is the shortest curve from 0 to P? What is its length? What is the distance d B 2(0, P )? c. Show that the ball B db 2 (0, r) in B 2 coincides with the euclidean open disk of radius tanh r 2 = e r 2 e r 2 e r 2 +e r 2 centered at 0. Problem 2. We now consider the hyperbolic plane (H 2, d hyp ), and the isometry Φ: H 2 B 2 from (H 2, d hyp ) to (B 2, d B 2) defined by Φ(z) = z + i z + i. Also consider the linear fractional map Ψ defined by Ψ(z) = iz + i z + 1. a. Show that Φ Ψ(z) = Ψ Φ(z) = z for every z. Conclude that Ψ sends every point of B 2 to a point of H 2, and defines an isometry from (B 2, d B 2) to (H 2, d hyp ). b. Use Part a to show that, for the point i = Ψ(0), the ball B dhyp (i, r) is the image of the ball B db 2 (0, r) under Ψ, namely that B dhyp (i, r) = Ψ ( B db 2 (0, r) ). c. Use Problem 1 and a certain property of linear fractional maps to show that B dhyp (i, r) is bounded by a (euclidean) circle C. d. Show that Ψ sends the x axis to the y axis. Conclude that the circle C contains the points Ψ(tanh r 2 ) = e r, Ψ( tanh r 2 ) = er, and is orthogonal to the y axis. e. Show that the ball B dhyp (i, r) is the open euclidean disk whose euclidean center is i cosh r (not i!) and whose euclidean radius is sinh r. Just in case, I remind you that cosh r = er +e r 2 and sinh r = er e r 2. f. Show that, for every y > 0, the ball B dhyp (iy, r) is an open euclidean disk. What are its euclidean center and euclidean radius? (Hint: homothety.)

9 g. Show that, for every z = x+iy H 2, the ball B dhyp (z, r) is an open euclidean disk. What are its euclidean center and euclidean radius? (Hint: horizontal translation.) Problem 3. In the sphere S 2, let N = (0, 0, 1) be the North Pole. Describe each of the balls B dsph (N, π 2 ), B d sph (N, π), B dsph (N, 3π 2 ) and B d sph (N, 2π) with a picture and a few words.

10 Homework 7, due on Wednesday October 26 Problem 1. Remember that the metric space (X, d) is locally isometric to the metric space (X, d ) if, for every P X, there exists an isometry ϕ: B d (P, r) B d (P, r) from a small ball B d (P, r) centered at P in X and a small ball B d (P, r) in X. Also, (X, d) is locally homogeneous if, for every P, Q X, there exists an isometry ϕ: B d (P, r) B d (Q, r) from a small ball B d (P, r) centered at P in X to a small ball B d (Q, r) centered at Q. Show that, if (X, d) is locally isometric to (X, d ) and if (X, d ) is locally homogeneous, then (X, d) is locally homogeneous. Problem 2. In the plane X = R 2, consider for each c R the hyperbola H c = {(x, y) R 2 ; xy = c}. (When c = 0, the hyperbola H 0 is somewhat degenerate.) a. Draw a picture of H 1, H 1, H 0 and H 1. 2 b. Show that the hyperbolas H c form a partition X of X = R 2, in the sense that every point P R 2 belongs to one and only one hyperbola H c. c. Consider the hyperbolas H c1 and H c2 associated to positive numbers c 1, c 2 > 0. Show that, for every ε > 0, there exist two points P 1 H c1 and P 2 H c2 such that d euc (P 1, P 2 ) < ε. d. More generally, consider the hyperbolas H c1 and H c2 associated to arbitrary numbers c 1, c 2 R. Show that, for every ε > 0, there exist two points P 1 H c1 and P 2 H c2 such that d euc (P 1, P 2 ) < ε. e. Let d euc be the quotient semi-metric on the partition X defined (using discrete walks as seen in class) by the euclidean metric d euc of X = R 2. In particular, for P R 2, let P X denote the hyperbola H c that contains it. (i) Show that d euc ( P 1, P 2 ) d euc (P 1, P 2 ), for every P 1 H c1 and P 2 H c2. (Hint: Can you find a discrete walk from P 1 to P 2?) (ii) Conclude that d euc ( P 1, P 2 ) = 0 for every P 1, P2 X. Hint: Part d. (iii) Is ( X, d euc ) a metric space?

11 Homework 8, due on Wednesday November 3 Recall from class that a homeomorphism from the metric space (X, d) to the metric space (X, d ) is a bijective map ϕ: X X such that both ϕ and its inverse ϕ 1 are continuous. Problem 1. Let X be a regular decagon (= polygon with 10 edges and 10 vertices) in the euclidean plane (R 2, d euc ), and let ( X, d euc ) be the quotient space obtained by gluing by euclidean translations opposite edges of the decagon X. a. The vertices of X correspond to how many points of X? b. Is the quotient space ( X, d euc ) locally isometric to the euclidean plane (R 2, d euc )? Explain. c. Give a proof by pictures, like the ones we have used in class in recent weeks, suggesting that the quotient space ( X, d euc ) is homeomorphic to the surface of genus 2 (namely the surface we already obtained by gluing opposite edges of an octagon). d. (No credit) If we glue opposite sides of a 2n gon X in R 2, what do you think the quotient space X is homeomorphic to? (Hint: do you see a pattern in the cases n = 2, 3, 4, 5? Problem 2. Let X be the square {(x, y) R 2 ; 0 x 1, 0 y 1} in the euclidean plane, and let ( X, d euc ) be the Klein bottle obtained from X by gluing each point (0, y) to the point (1, y), and each point (x, 0) to (1 x, 1). Draw a picture of X and indicate by arrows the gluing of its sides, as we have done in class. a. Let α be the horizontal line segment {(x, y) X; y = 1 2 } in X. Draw a picture of α. Show that its image ᾱ in X is a closed curve, namely that its end points are glued together. b. For ᾱ as in Part a, let X ᾱ consists of all points P X that are not in ᾱ. What is X ᾱ homeomorphic to? (Use a proof by picture.) c. Let β be the vertical line segment {(x, y) X; x = 1 2 } in X. Draw a picture of β. Show that its image β is a closed curve in X. What is X β homeomorphic to? (Use a proof by picture.) d. Let γ consist of the two vertical line segments {(x, y) X; x = 1 3 or 2 3 } in X. Draw a picture of γ. Show that the image γ consists of a single closed curve in X. What is X β homeomorphic to? (Use a proof by picture.)

12 , Practice Final Exam Problem 1. (10%) Sketch a tessellation of the euclidean plane R 2 by triangles whose angles are π 2, π 3, π 6. Problem 2. Let X be a regular dodecagon in the hyperbolic plane (H 2, d hyp ), with all 12 sides of equal lengths and all 12 angles equal to θ. Label the vertices of X as V 1, V 2,..., V 12 in this order around X, and glue the edge A 1 A 2 to A 8 A 7, the edge A 2 A 3 to A 1 A 12, the edge A 3 A 4 to A 6 A 5, the edge A 4 A 5 to A 11 A 10, the edge A 6 A 7 to A 9 A 8, and the edge A 9 A 10 to A 12 A 11. (It may help to draw arrows on the picture below, which represents X is the disk model for symmetry.) Let ( X, d hyp ) be the corresponding quotient space. A 12 A 1 A 11 A 2 A 10 A 3 A 9 A 4 A 8 A 5 A 7 a. (8%) How many points of X correspond to the vertices of X? b. (8%) For which value of θ is the quotient space ( X, d hyp ) locally isometric to the hyperbolic plane (H 2, d hyp )? A 6 Problem 3. (12%) Let X be a polygon in the euclidean plane R 2, and let X be the quotient space obtained by gluing edges of X together. Given two points P, Q X in this quotient space, give the precise definition of a discrete walk w from P to Q, and of the length l d (w) of this discrete walk. Problem 4. We want to endow the real line R with a new metric d, defined by the property that 0 if x = y d(x, y) = max{ 1 q ; p, q integers, q > 0, x < p q < y} if x < y max{ 1 q ; p, q integers, q > 0, y < p q < x} if x > y. (Namely, d(x, y) is 1 over the smallest denominator of a rational number sitting between x and y.) a. (6%) Compute d(0, 1 2 ), d(0, 1 3 ) and d( 1 2, 1 3 ).

13 b. (14%) Show that d is indeed a metric, and that (R, d) is a metric space. Problem 5. Consider a hyperbolic isometry ϕ(z) = az + b with a, b, c, d R, ad bc = 1 and c 0 cz + d and the horizontal line L = {x + i; x R} defined by the equation y = 1. a. (4%) Compute ϕ( ). b. (8%) Remember that we saw in class that a linear fractional map sends circle to circle (if we consider a line plus the point as a circle of infinite radius). Use this property to show that ϕ sends L to a C { a c }, where C is a circle in C that is tangent to the real line R at the point a c, and where C { a c } denotes the circle C from which the point a c has been removed. c. (continuation of Problem 5) (10%) Compute the imaginary part of ϕ(x + i), and find the maximum of this imaginary part as x ranges over all points of R. d. (2%) Use Part b to find the radius of the circle C. Problem 6. (18%) Let ϕ: H 2 H 2 be a hyperbolic isometry sending the point to itself. Show that, either ϕ is a horizontal translation ϕ(z) = z + x 0 with x 0 R, or there exists a complete geodesic g such that ϕ(g) = g. (Possible hint: Write ϕ as ϕ(z) = az+b cz+d or ϕ(z) = c z+d a z+b and look for the end points of g.)

Notes for MATH 434 Geometry and Transformations. Francis Bonahon Fall 2015

Notes for MATH 434 Geometry and Transformations. Francis Bonahon Fall 2015 Notes for MATH 434 Geometry and Transformations Francis Bonahon Fall 2015 Version: October 21, 2016 Department of Mathematics University of Southern California Los Angeles, CA 90089-2532, U.S.A. E-mail

More information

DIFFERENTIAL GEOMETRY HW 5

DIFFERENTIAL GEOMETRY HW 5 DIFFERENTIAL GEOMETRY HW 5 CLAY SHONKWILER 1 Check the calculations above that the Gaussian curvature of the upper half-plane and Poincaré disk models of the hyperbolic plane is 1. Proof. The calculations

More information

Part IB GEOMETRY (Lent 2016): Example Sheet 1

Part IB GEOMETRY (Lent 2016): Example Sheet 1 Part IB GEOMETRY (Lent 2016): Example Sheet 1 (a.g.kovalev@dpmms.cam.ac.uk) 1. Suppose that H is a hyperplane in Euclidean n-space R n defined by u x = c for some unit vector u and constant c. The reflection

More information

Part IB Geometry. Theorems. Based on lectures by A. G. Kovalev Notes taken by Dexter Chua. Lent 2016

Part IB Geometry. Theorems. Based on lectures by A. G. Kovalev Notes taken by Dexter Chua. Lent 2016 Part IB Geometry Theorems Based on lectures by A. G. Kovalev Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.

More information

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

More information

Möbius Transformation

Möbius Transformation Möbius Transformation 1 1 June 15th, 2010 Mathematics Science Center Tsinghua University Philosophy Rigidity Conformal mappings have rigidity. The diffeomorphism group is of infinite dimension in general.

More information

5.3 The Upper Half Plane

5.3 The Upper Half Plane Remark. Combining Schwarz Lemma with the map g α, we can obtain some inequalities of analytic maps f : D D. For example, if z D and w = f(z) D, then the composition h := g w f g z satisfies the condition

More information

2 hours THE UNIVERSITY OF MANCHESTER.?? January 2017??:????:??

2 hours THE UNIVERSITY OF MANCHESTER.?? January 2017??:????:?? hours MATH3051 THE UNIVERSITY OF MANCHESTER HYPERBOLIC GEOMETRY?? January 017??:????:?? Answer ALL FOUR questions in Section A (40 marks in all) and TWO of the THREE questions in Section B (30 marks each).

More information

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2. 1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

More information

Hyperbolic Transformations

Hyperbolic Transformations C H A P T E R 17 Hyperbolic Transformations Though the text of your article on Crystal Symmetry and Its Generalizations is much too learned for a simple, selfmade pattern man like me, some of the text-illustrations

More information

Math 417 Midterm Exam Solutions Friday, July 9, 2010

Math 417 Midterm Exam Solutions Friday, July 9, 2010 Math 417 Midterm Exam Solutions Friday, July 9, 010 Solve any 4 of Problems 1 6 and 1 of Problems 7 8. Write your solutions in the booklet provided. If you attempt more than 5 problems, you must clearly

More information

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain. Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

More information

Hyperbolic Geometry on Geometric Surfaces

Hyperbolic Geometry on Geometric Surfaces Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction

More information

MATH 162. Midterm 2 ANSWERS November 18, 2005

MATH 162. Midterm 2 ANSWERS November 18, 2005 MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able

More information

612 CLASS LECTURE: HYPERBOLIC GEOMETRY

612 CLASS LECTURE: HYPERBOLIC GEOMETRY 612 CLASS LECTURE: HYPERBOLIC GEOMETRY JOSHUA P. BOWMAN 1. Conformal metrics As a vector space, C has a canonical norm, the same as the standard R 2 norm. Denote this dz one should think of dz as the identity

More information

Solutions to Exercises 6.1

Solutions to Exercises 6.1 34 Chapter 6 Conformal Mappings Solutions to Exercises 6.. An analytic function fz is conformal where f z. If fz = z + e z, then f z =e z z + z. We have f z = z z += z =. Thus f is conformal at all z.

More information

Highly complex: Möbius transformations, hyperbolic tessellations and pearl fractals

Highly complex: Möbius transformations, hyperbolic tessellations and pearl fractals Highly complex: Möbius transformations, hyperbolic tessellations and pearl fractals Department of mathematical sciences Aalborg University Cergy-Pontoise 26.5.2011 Möbius transformations Definition Möbius

More information

(7) Suppose α, β, γ are nonzero complex numbers such that α = β = γ.

(7) Suppose α, β, γ are nonzero complex numbers such that α = β = γ. January 22, 2011 COMPLEX ANALYSIS: PROBLEMS SHEET -1 M.THAMBAN NAIR (1) Show that C is a field under the addition and multiplication defined for complex numbers. (2) Show that the map f : R C defined by

More information

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial. Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and

More information

Second Midterm Exam Name: Practice Problems March 10, 2015

Second Midterm Exam Name: Practice Problems March 10, 2015 Math 160 1. Treibergs Second Midterm Exam Name: Practice Problems March 10, 015 1. Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z

More information

Conformal Mappings. Chapter Schwarz Lemma

Conformal Mappings. Chapter Schwarz Lemma Chapter 5 Conformal Mappings In this chapter we study analytic isomorphisms. An analytic isomorphism is also called a conformal map. We say that f is an analytic isomorphism of U with V if f is an analytic

More information

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012 INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012 September 5, 2012 Mapping Properties Lecture 13 We shall once again return to the study of general behaviour of holomorphic functions

More information

Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:

Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph: Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates

More information

PICARD S THEOREM STEFAN FRIEDL

PICARD S THEOREM STEFAN FRIEDL PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A

More information

IV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations

IV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 IV. Conformal Maps 1. Geometric interpretation of differentiability We saw from the definition of complex differentiability that if f

More information

Plane hyperbolic geometry

Plane hyperbolic geometry 2 Plane hyperbolic geometry In this chapter we will see that the unit disc D has a natural geometry, known as plane hyperbolic geometry or plane Lobachevski geometry. It is the local model for the hyperbolic

More information

Math 814 HW 3. October 16, p. 54: 9, 14, 18, 24, 25, 26

Math 814 HW 3. October 16, p. 54: 9, 14, 18, 24, 25, 26 Math 814 HW 3 October 16, 2007 p. 54: 9, 14, 18, 24, 25, 26 p.54, Exercise 9. If T z = az+b, find necessary and sufficient conditions for T to cz+d preserve the unit circle. T preserves the unit circle

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 24, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 24, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 24, 2011 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

More information

AP CALCULUS AB Study Guide for Midterm Exam 2017

AP CALCULUS AB Study Guide for Midterm Exam 2017 AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

Part II. Geometry and Groups. Year

Part II. Geometry and Groups. Year Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),

More information

THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES

THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES IGOR PROKHORENKOV 1. Introduction to the Selberg Trace Formula This is a talk about the paper H. P. McKean: Selberg s Trace Formula as applied to a

More information

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE

THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE THE AUTOMORPHISM GROUP ON THE RIEMANN SPHERE YONG JAE KIM Abstract. In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space

More information

274 Curves on Surfaces, Lecture 4

274 Curves on Surfaces, Lecture 4 274 Curves on Surfaces, Lecture 4 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 4 Hyperbolic geometry Last time there was an exercise asking for braids giving the torsion elements in PSL 2 (Z). A 3-torsion

More information

Analysis-3 lecture schemes

Analysis-3 lecture schemes Analysis-3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space

More information

Inleiding Topologie. Lecture Notes. Marius Crainic

Inleiding Topologie. Lecture Notes. Marius Crainic Inleiding Topologie Lecture Notes Marius Crainic c Mathematisch Instituut Universiteit Utrecht Aangepast, November 2017 Contents Chapter 1. Introduction: some standard spaces 5 1. Keywords for this course

More information

Introduction to Real Analysis

Introduction to Real Analysis Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that

More information

A crash course the geometry of hyperbolic surfaces

A crash course the geometry of hyperbolic surfaces Lecture 7 A crash course the geometry of hyperbolic surfaces 7.1 The hyperbolic plane Hyperbolic geometry originally developed in the early 19 th century to prove that the parallel postulate in Euclidean

More information

GEOMETRY Notes Easter 2002

GEOMETRY Notes Easter 2002 Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY Notes Easter 2002 T. K. Carne. t.k.carne@dpmms.cam.ac.uk c Copyright. Not for distribution outside Cambridge

More information

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017 Math 2 Lesieutre; 9: Polar coordinates; November 22, 207 Plot the point 2, 2 in the plane If you were trying to describe this point to a friend, how could you do it? One option would be coordinates, but

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

MATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE

MATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE MATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE 1. Introduction Let D denote the unit disk and let D denote its boundary circle. Consider a piecewise continuous function on the boundary circle, {

More information

Lecture Figure 4.5. Relating curvature to the circumference of a circle.

Lecture Figure 4.5. Relating curvature to the circumference of a circle. Lecture 26 181 Figure 4.5. Relating curvature to the circumference of a circle. the plane with radius r (Figure 4.5). We will see that circumference = 2πr cr 3 + o(r 3 ) where c is a constant related to

More information

z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1

z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1 3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

More information

Since x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)

Since x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C) SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2

More information

Chapter 14 Hyperbolic geometry Math 4520, Fall 2017

Chapter 14 Hyperbolic geometry Math 4520, Fall 2017 Chapter 14 Hyperbolic geometry Math 4520, Fall 2017 So far we have talked mostly about the incidence structure of points, lines and circles. But geometry is concerned about the metric, the way things are

More information

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables Lecture Complex Numbers MATH-GA 245.00 Complex Variables The field of complex numbers. Arithmetic operations The field C of complex numbers is obtained by adjoining the imaginary unit i to the field R

More information

Friday 09/15/2017 Midterm I 50 minutes

Friday 09/15/2017 Midterm I 50 minutes Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.

More information

10.1 Curves Defined by Parametric Equation

10.1 Curves Defined by Parametric Equation 10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical

More information

(x, y) = d(x, y) = x y.

(x, y) = d(x, y) = x y. 1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance

More information

MAT389 Fall 2016, Problem Set 2

MAT389 Fall 2016, Problem Set 2 MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)

More information

Fourth Week: Lectures 10-12

Fourth Week: Lectures 10-12 Fourth Week: Lectures 10-12 Lecture 10 The fact that a power series p of positive radius of convergence defines a function inside its disc of convergence via substitution is something that we cannot ignore

More information

MAT1035 Analytic Geometry

MAT1035 Analytic Geometry MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................

More information

III.3. Analytic Functions as Mapping, Möbius Transformations

III.3. Analytic Functions as Mapping, Möbius Transformations III.3. Analytic Functions as Mapping, Möbius Transformations 1 III.3. Analytic Functions as Mapping, Möbius Transformations Note. To graph y = f(x) where x,y R, we can simply plot points (x,y) in R 2 (that

More information

Algorithms for Picture Analysis. Lecture 07: Metrics. Axioms of a Metric

Algorithms for Picture Analysis. Lecture 07: Metrics. Axioms of a Metric Axioms of a Metric Picture analysis always assumes that pictures are defined in coordinates, and we apply the Euclidean metric as the golden standard for distance (or derived, such as area) measurements.

More information

Solution Sheet 1.4 Questions 26-31

Solution Sheet 1.4 Questions 26-31 Solution Sheet 1.4 Questions 26-31 26. Using the Limit Rules evaluate i) ii) iii) 3 2 +4+1 0 2 +4+3, 3 2 +4+1 2 +4+3, 3 2 +4+1 1 2 +4+3. Note When using a Limit Rule you must write down which Rule you

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 25, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 25, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 25, 2010 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

More information

THE VOLUME OF A HYPERBOLIC 3-MANIFOLD WITH BETTI NUMBER 2. Marc Culler and Peter B. Shalen. University of Illinois at Chicago

THE VOLUME OF A HYPERBOLIC 3-MANIFOLD WITH BETTI NUMBER 2. Marc Culler and Peter B. Shalen. University of Illinois at Chicago THE VOLUME OF A HYPERBOLIC -MANIFOLD WITH BETTI NUMBER 2 Marc Culler and Peter B. Shalen University of Illinois at Chicago Abstract. If M is a closed orientable hyperbolic -manifold with first Betti number

More information

MATH 103 Pre-Calculus Mathematics Dr. McCloskey Fall 2008 Final Exam Sample Solutions

MATH 103 Pre-Calculus Mathematics Dr. McCloskey Fall 2008 Final Exam Sample Solutions MATH 103 Pre-Calculus Mathematics Dr. McCloskey Fall 008 Final Exam Sample Solutions In these solutions, FD refers to the course textbook (PreCalculus (4th edition), by Faires and DeFranza, published by

More information

We introduce the third of the classical geometries, hyperbolic geometry.

We introduce the third of the classical geometries, hyperbolic geometry. Chapter Hyperbolic Geometry We introduce the third of the classical geometries, hyperbolic geometry.. Hyperbolic Geometry Lines are (i) vertical lines in H H = H 2 = {(x, y) R 2 y > 0} = {z C Im(z) > 0}

More information

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

More information

Welcome to AP Calculus!!!

Welcome to AP Calculus!!! Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you

More information

DIFFERENTIAL GEOMETRY HW 5. Show that the law of cosines in spherical geometry is. cos c = cos a cos b + sin a sin b cos θ.

DIFFERENTIAL GEOMETRY HW 5. Show that the law of cosines in spherical geometry is. cos c = cos a cos b + sin a sin b cos θ. DIFFEENTIAL GEOMETY HW 5 CLAY SHONKWILE Show that the law of cosines in spherical geometry is 5 cos c cos a cos b + sin a sin b cos θ. Proof. Consider the spherical triangle depicted below: Form radii

More information

1 Euclidean geometry. 1.1 The metric on R n

1 Euclidean geometry. 1.1 The metric on R n 1 Euclidean geometry This chapter discusses the geometry of n-dimensional Euclidean space E n, together with its distance function. The distance gives rise to other notions such as angles and congruent

More information

Lecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables

Lecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables Lecture 14 Conformal Mapping MATH-GA 2451.001 Complex Variables 1 Conformality 1.1 Preservation of angle The open mapping theorem tells us that an analytic function such that f (z 0 ) 0 maps a small neighborhood

More information

13 Spherical geometry

13 Spherical geometry 13 Spherical geometry Let ABC be a triangle in the Euclidean plane. From now on, we indicate the interior angles A = CAB, B = ABC, C = BCA at the vertices merely by A, B, C. The sides of length a = BC

More information

Thus, X is connected by Problem 4. Case 3: X = (a, b]. This case is analogous to Case 2. Case 4: X = (a, b). Choose ε < b a

Thus, X is connected by Problem 4. Case 3: X = (a, b]. This case is analogous to Case 2. Case 4: X = (a, b). Choose ε < b a Solutions to Homework #6 1. Complete the proof of the backwards direction of Theorem 12.2 from class (which asserts the any interval in R is connected). Solution: Let X R be a closed interval. Case 1:

More information

Complex Analysis Homework 1: Solutions

Complex Analysis Homework 1: Solutions Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4

More information

MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm.

MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper

More information

William P. Thurston. The Geometry and Topology of Three-Manifolds

William P. Thurston. The Geometry and Topology of Three-Manifolds William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed

More information

Chapter 3: Inequalities, Lines and Circles, Introduction to Functions

Chapter 3: Inequalities, Lines and Circles, Introduction to Functions QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used

More information

Math 461 Homework 8. Paul Hacking. November 27, 2018

Math 461 Homework 8. Paul Hacking. November 27, 2018 Math 461 Homework 8 Paul Hacking November 27, 2018 (1) Let S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F :

More information

Complex Practice Exam 1

Complex Practice Exam 1 Complex Practice Exam This practice exam contains sample questions. The actual exam will have fewer questions, and may contain questions not listed here.. Be prepared to explain the following concepts,

More information

Module 2: Reflecting on One s Problems

Module 2: Reflecting on One s Problems MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations

More information

Poincaré Models of Hyperbolic Geometry

Poincaré Models of Hyperbolic Geometry Chapter 9 Poincaré Models of Hyperbolic Geometry 9.1 The Poincaré Upper Half Plane Model The next model of the hyperbolic plane that we will consider is also due to Henri Poincaré. We will be using the

More information

PreCalculus Honors Curriculum Pacing Guide First Half of Semester

PreCalculus Honors Curriculum Pacing Guide First Half of Semester Unit 1 Introduction to Trigonometry (9 days) First Half of PC.FT.1 PC.FT.2 PC.FT.2a PC.FT.2b PC.FT.3 PC.FT.4 PC.FT.8 PC.GCI.5 Understand that the radian measure of an angle is the length of the arc on

More information

INTRODUCTION TO GEOMETRY

INTRODUCTION TO GEOMETRY INTRODUCTION TO GEOMETRY ERIKA DUNN-WEISS Abstract. This paper is an introduction to Riemannian and semi-riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames,

More information

2 Recollection of elementary functions. II

2 Recollection of elementary functions. II Recollection of elementary functions. II Last updated: October 5, 08. In this section we continue recollection of elementary functions. In particular, we consider exponential, trigonometric and hyperbolic

More information

2009 Math Olympics Level II Solutions

2009 Math Olympics Level II Solutions Saginaw Valley State University 009 Math Olympics Level II Solutions 1. f (x) is a degree three monic polynomial (leading coefficient is 1) such that f (0) 3, f (1) 5 and f () 11. What is f (5)? (a) 7

More information

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS

MATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if

More information

Physical Chemistry - Problem Drill 02: Math Review for Physical Chemistry

Physical Chemistry - Problem Drill 02: Math Review for Physical Chemistry Physical Chemistry - Problem Drill 02: Math Review for Physical Chemistry No. 1 of 10 1. The Common Logarithm is based on the powers of 10. Solve the logarithmic equation: log(x+2) log(x-1) = 1 (A) 1 (B)

More information

Math 421 Midterm 2 review questions

Math 421 Midterm 2 review questions Math 42 Midterm 2 review questions Paul Hacking November 7, 205 () Let U be an open set and f : U a continuous function. Let be a smooth curve contained in U, with endpoints α and β, oriented from α to

More information

MTH 362: Advanced Engineering Mathematics

MTH 362: Advanced Engineering Mathematics MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017 Course Name and number: MTH 362: Advanced Engineering

More information

What is a vector in hyperbolic geometry? And, what is a hyperbolic linear transformation?

What is a vector in hyperbolic geometry? And, what is a hyperbolic linear transformation? 0 What is a vector in hyperbolic geometry? And, what is a hyperbolic linear transformation? Ken Li, Dennis Merino, and Edgar N. Reyes Southeastern Louisiana University Hammond, LA USA 70402 1 Introduction

More information

INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES

INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES PHILIP FOTH 1. Cauchy s Formula and Cauchy s Theorem 1. Suppose that γ is a piecewise smooth positively ( counterclockwise ) oriented simple closed

More information

Math 147, Homework 1 Solutions Due: April 10, 2012

Math 147, Homework 1 Solutions Due: April 10, 2012 1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M

More information

Weekly Activities Ma 110

Weekly Activities Ma 110 Weekly Activities Ma 110 Fall 2008 As of October 27, 2008 We give detailed suggestions of what to learn during each week. This includes a reading assignment as well as a brief description of the main points

More information

MA424, S13 HW #6: Homework Problems 1. Answer the following, showing all work clearly and neatly. ONLY EXACT VALUES WILL BE ACCEPTED.

MA424, S13 HW #6: Homework Problems 1. Answer the following, showing all work clearly and neatly. ONLY EXACT VALUES WILL BE ACCEPTED. MA424, S13 HW #6: 44-47 Homework Problems 1 Answer the following, showing all work clearly and neatly. ONLY EXACT VALUES WILL BE ACCEPTED. NOTATION: Recall that C r (z) is the positively oriented circle

More information

Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.

Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder. Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the

More information

1. (4 % each, total 20 %) Answer each of the following. (No need to show your work for this problem). 3 n. n!? n=1

1. (4 % each, total 20 %) Answer each of the following. (No need to show your work for this problem). 3 n. n!? n=1 NAME: EXAM 4 - Math 56 SOlutions Instruction: Circle your answers and show all your work CLEARLY Partial credit will be given only when you present what belongs to part of a correct solution (4 % each,

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

1 z n = 1. 9.(Problem) Evaluate each of the following, that is, express each in standard Cartesian form x + iy. (2 i) 3. ( 1 + i. 2 i.

1 z n = 1. 9.(Problem) Evaluate each of the following, that is, express each in standard Cartesian form x + iy. (2 i) 3. ( 1 + i. 2 i. . 5(b). (Problem) Show that z n = z n and z n = z n for n =,,... (b) Use polar form, i.e. let z = re iθ, then z n = r n = z n. Note e iθ = cos θ + i sin θ =. 9.(Problem) Evaluate each of the following,

More information

INTEGRATION WORKSHOP 2003 COMPLEX ANALYSIS EXERCISES

INTEGRATION WORKSHOP 2003 COMPLEX ANALYSIS EXERCISES INTEGRATION WORKSHOP 23 COMPLEX ANALYSIS EXERCISES DOUGLAS ULMER 1. Meromorphic functions on the Riemann sphere It s often useful to allow functions to take the value. This exercise outlines one way to

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

Math 3c Solutions: Exam 2 Fall 2017

Math 3c Solutions: Exam 2 Fall 2017 Math 3c Solutions: Exam Fall 07. 0 points) The graph of a smooth vector-valued function is shown below except that your irresponsible teacher forgot to include the orientation!) Several points are indicated

More information

Mathematics Specialist Units 3 & 4 Program 2018

Mathematics Specialist Units 3 & 4 Program 2018 Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review

More information

MATH 32 FALL 2013 FINAL EXAM SOLUTIONS. 1 cos( 2. is in the first quadrant, so its sine is positive. Finally, csc( π 8 ) = 2 2.

MATH 32 FALL 2013 FINAL EXAM SOLUTIONS. 1 cos( 2. is in the first quadrant, so its sine is positive. Finally, csc( π 8 ) = 2 2. MATH FALL 01 FINAL EXAM SOLUTIONS (1) (1 points) Evalute the following (a) tan(0) Solution: tan(0) = 0. (b) csc( π 8 ) Solution: csc( π 8 ) = 1 sin( π 8 ) To find sin( π 8 ), we ll use the half angle formula:

More information

1. Complex Numbers. John Douglas Moore. July 1, 2011

1. Complex Numbers. John Douglas Moore. July 1, 2011 1. Complex Numbers John Douglas Moore July 1, 2011 These notes are intended to supplement the text, Fundamentals of complex analysis, by Saff and Snider [5]. Other often-used references for the theory

More information