Math 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech
|
|
- Jeffery Hunt
- 5 years ago
- Views:
Transcription
1 Mth 6455 Oct 10, Differentil Geometry I Fll 2006, Georgi Tech Lecture Notes 12 Riemnnin Metrics 0.1 Definition If M is smooth mnifold then by Riemnnin metric g on M we men smooth ssignment of n innerproduct to ech tngent spce of M. This mens tht, for ech p M, g p : T p M T p M R is symmetric, positive definite, biliner mp, nd furthermore the ssignment p g p is smooth, i.e., for ny smooth vector fields X nd Y on M, p g p (X p, Y p ) is smooth function. The pir (M, g) then will be clled Riemnnin mnifold. We sy tht diffeomorphism f : M N between pir of Riemnnin mnifolds (M, g) nd (N, h) is n isometry provided tht for ll p M nd X, Y T p M. g p (X, Y ) = h f(p) (df p (X), df p (Y )) Exercise Show tht the ntipodl reflection : S n S n, (x) := x is n isometry. 0.2 Exmples The Eucliden Spce The simplest exmple of Riemnnin mnifold is R n with its stndrd Eucliden innerproduct, g(x, Y ) := X, Y Submnifolds of Riemnnin mnifold A rich source of exmples re generted by immersions f : N M of ny mnifold N into Riemnnin mnifold M (with metric g); for this induces metric h on N given by h p (X, Y ) := g f(p) (df p (X), df p (Y )). In prticulr ny mnifold my be equipped with Riemnnin metric since every mnifold dmits n embedding into R n. 1 Lst revised: November 23,
2 0.2.3 Quotient of Riemnnin mnifold by group of isometries Note tht the set of isometries f : M M forms group. Another source of exmples of Riemnnin mnifolds re generted by tking the quotient of Riemnnin mnifold (M, g) by subgroup G of its isometries which cts properly discontinuously on M. Recll tht if G cts properly discontinuously, then M/G is indeed mnifold. Then we my define metric h on M/G by setting h [p] := g p. More precisely recll tht the projections π : M M/G, given by π(p) := [p] is locl diffeomorphism, i.e., for ny q [p] there exists n open neighborhood U of p in M nd n open neighborhood V of [p] in M/G such tht π : U V is diffeomorphism. Then we my define h [p] (X, Y ) := g q ((dπ q ) 1 (X), (dπ q ) 1 (Y )). One cn immeditely check tht h does not depend on the choice of q [p] nd is thus well defined.a specific exmple of proper discontinuous ction of isometries is given by trnsltions f z : R n R n given by f z (p) := p + z where z Z n. Recll tht R n /Z n is the torus T n, which my now be equipped with the metric induced by this group ction. Similrly RP n dmits cnonicl metric, since RP n = S n /{±1}, nd reflections of sphere re isometries Conforml trnsformtions As nother set of exmples note tht if (M, g) is ny Riemnnin mnifold, then (M, λg) is lso Riemnnin mnifold where λ: M R + is ny smooth positive function. Note tht this chnge of metric does not effect the ngles between ny pir of vectors in tngent spce of M. Thus (M, λg) is sid to be conforml to (M, g). Exercise Show tht the inversion i: R n {o} R n given by i(x) := x/ x is conforml trnsformtion. Exercise Show tht the stereogrphic projection π : S 2 {(0, 0, 1)} R 2 is conforml trnsformtion The hyperbolic spce Finlly, n importnt exmple is the hyperbolic spce which my be represented by number of models. One model, known s Poincre s hlf spce model, is to tke the open upper hlf spce of R n nd define there metric vi g p (X, Y ) := X, Y (p n ) 2, 2
3 where p n denotes the n th coordinte of p. Another description of the hyperbolic spce my be given by tking the open unit bll if R n nd defining This is known s Poincre s bll model. g p (X, Y ) := X, Y (1 p 2 ) 2. Exercise Show tht the the Poincre hlf-plne nd the hlf-disk re isometric (Hint: identify the Poincre hlf-plne with the region y > 1 in R 2 nd do n inversion). 0.3 Metric in locl coordintes Let (U, φ) be locl chrt for (M, g). Then, recll tht if e 1,... e n denote the stndrd bsis of R n, we obtin bsis for ech T p M, for p U by setting E i (p) := dφ 1 φ(p) (e i). Now if X, Y T p M, then X = n i=1 Xi E i nd Y = n i=1 Y i E i. Further, if we set g ij (p) := g p (E i, E j ), then, since g is biliner we hve g p (X, Y ) = X i Y j g p (E i, E j ) = X i Y j g ij (p). Thus in ny locl coordinte (U, φ) metric is completely determined by the functions g ij which my be regrded s the coefficients of positive definite mtrix. To obtin concrete exmple, note tht if M R n is submnifold, with the induce metric from R n, nd (φ, U) is locl chrt of M, then if we set f := φ 1, f : φ(u) R n is prmetriztion for U, nd d(f)(e i ) = D i f. Consequently, g ij (p) = D i f(f 1 (p)), D j f(f 1 (p)). For instnce, note tht surfce of revolution in R 3 which is given by rotting the curve (r(t), z(t)) in the xz-plne bout the z xis cn be prmetrized by So f(t, θ) = (r(t) cos θ, r(t) sin θ, z(t)). D 1 f(t, θ) = (r (t) cos θ, r (t) sin θ, z (t)) nd D 2 f(t, θ) = ( r(t) sin θ, r(t) cos θ, 0), nd consequently g ij (f(t, θ)) is given by ( (r ) 2 + (z ) r 2 Note tht if we ssume tht the curve in the xz-plne is prmetrized by rclength, then (r ) 2 + (z ) 2 = 1, so the bove mtrix becomes more simple to work with. 3 ).
4 Exercise Compute the metric of S 2 in terms of sphericl coordintes θ nd φ. Exercise Compute the metric of the surfce given by the grph of function f : Ω R 2 R. 0.4 Length of Curves In Riemnnin mnifold (M, g), the length of ny piecewise smooth curves c: [, b] M with c() = p nd c(b) = q is defined s where Length[c] := g c(t) (c (t), c (t)) dt, c (t) := dc t (1). Note tht the definition for the length of curves here is generliztion of the Eucliden cse where we integrte the speed of the curve. Indeed the lst formul bove coincides with the regulr notion of derivtive when M is just R n. To see this, recll tht dc t (1) = (c γ) (0) where γ : (ɛ, ɛ) [, b] is curve with γ(0) = t nd γ (0) = 1, e.g., γ(u) = t+u. Thus by the chin rule (c γ) (0) = c (γ(0))γ (0) = c (t). Exercise Compute the length of the rdius of the Poincre-disk (with respect to the Poincre metric). 0.5 The clssicl nottion for metric For ny curve c: [, b] R n we my write c(t) = (x 1 (t),..., x n (t)). Consequently, if we define g ij (p) := g p (e i, e j ) where e 1,..., e n is the stndrd bsis for R n, then bilinerity of g yields tht g c(t) (c (t), c (t)) = g c(t) (e i, e j )x i(t)x j(t) = g ij (c(t))x i(t)x j(t). Thus we my write Length[c] := g ij (c(t)) dx i dt dx j dt dt. Indeed clssiclly metrics were specified by n expression of the form ds 2 = g ij dx i dx j. 4
5 nd then length of curve ws defined s the integrl of ds, which ws clled the element of rclength, long tht curve: Length[c] = ds. In prticulr note tht, in the clssicl nottion, the stndrd Eucliden metric in the plne is given by ds 2 = n i=1 dx2 i. Further, in the Poincre s hlf-disk model, ds 2 = n i=1 dx2 i /x2 n. 0.6 Distnce For ny pirs of points p, q M, let C(p, q) denote the spce piecewise smooth curves c: [, b] M with c() = p nd c(b) = q. Then, if M is connected, we my define the distnce between p nd q s d g (p, q) := inf{length[c] c C(p, q)}. So the distnce between pir of points is defined s the gretest lower bound of the lengths of curves which connect those points. First we show tht this is generliztion of the stndrd notion of distnce in R n. Lemm For ll continuous mps f : (, b) R n f(t)dt f(t) dt. Proof. By the Cuchy-Schwrts inequlity, for ny unit vector u S n 1, f(t)dt, u = f(t), u dt f(t) dt. In prticulr we my let u := f(t)dt/ f(t)dt, ssuming tht f(t)dt 0 (otherwise the lemm is obviously true). Corollry If (M, g) = (R n, ) then d g (p, q) = p q. Proof. First note tht if we set c(t) := (1 t)p + tq, then Length[c] := 1 0 c p q dt = p q. So d g (p, q) p q. It remins then to show tht d g (p, q) p q. The lter inequlity holds becuse for ll curves c: [, b] R n c b (t) dt c (t)dt = c(b) c(). 5
6 The previous result shows tht (M, d g ) is metric spce when M is the Eucliden spce R n nd g, which induces d, is the stndrd innerproduct. Next we show tht this is the cse for ll Riemnnin mnifolds. To this end we first need locl lemm: Lemm Let (B, g) be Riemnnin mnifold, where B := B n r (o) R n. Then there exists m > 0 such tht for ny piecewise C 1 curve c: [, b] B with c() = o nd c(b) B we hve Length[c] > m. Proof. Define f : S n 1 B R by f(u, p) := g p (u, u). Note tht, since g is positive definite, f > 0. Thus since f is continuous nd S n 1 B is compct f λ 2 > 0. Consequently, bilinerity of g yields tht g p (v, v) λ 2 v 2. The bove inequlity is obvious when v = 0, nd when v 0, observe tht g p (v, v) = g p (v/ v, v/ v ) v 2. Next note tht Length[c] = g c(t) (c (t), c (t)) dt λ c (t) dt. But c (t) is just the length of c with respect to the stndrd metric on R n. Thus, by the previous proposition, So setting m := λr finishes the proof. c (t) dt c(b) c() = r. The proof of the next observtion is immedite: Lemm If f : M N is n isometry, then Length[c] = Length[f c] for ny piecewise C 1 curve c: [, b] M. Note tht if (M, g) is Riemnnin mnifold nd f : M N is diffeomorphism between M nd ny smooth mnifold N, then we my push forwrd the metric of M by defining df(g) p (X, Y ) := g f 1 (p)(df 1 (X), df 1 (Y )). Then f is n isometry between (M, g) nd (N, df(g)). In prticulr we my ssume tht ny locl chrts (U, φ) on Riemnnin mnifold (M, g) is n isometry, with respect to the push forwrd metric dφ(g) on φ(u). This observtion, together with the previous lemm esily yields tht: Proposition If (M, g) is ny Riemnnin mnifold then (M, d g ) is metric spce. 6
7 Proof. It is immedite tht d is symmetric nd stisfies the tringle inequlity. Furthermore it is cler tht d is lwys nonnegtive. Showing tht d is positive definite, however, requires more work. Specificlly, we need to show tht when p q, then d(p, q) > 0. Suppose p q. Then, since M is Husdorff, there exists n open neighborhood V of p such tht q V. Let (U, φ) be locl chrt centered t p. Choose r so smll tht B r (o) φ(v U), nd set W := φ 1 (B r (o)). Then φ: W B r (o) is diffeomorphism, nd we my equip B r (o) with the push forwrd metric dφ(g) which will turn φ into n isometry. Now let c: [, b] M be ny piecewise C 1 curve with c() = p nd c(b) = q. Then there exist b b such tht c[, b ] W nd c(b ) W (to find b let W := φ 1 (B r (o)) be the interior of W, then c 1 ( W ) is n open subset of [, b] which contins, nd we my let b be the upperbound of the component of c 1 ( W ) which contins.) Let c: [, b ] W be the restriction of c. Then obviously Length[c] Length[c]. But Length[c] = Length[φ c] since φ is n isometry, nd by the previous lemm then length of ny curve in (B n r (o), dφ(g)) which begins t the center of the bll nd ends t its boundry is bounded below by positive constnt. Now recll tht ny metric spce hs nturl topology. In prticulr (M, d g ) is topologicl spce. Next we show tht this topologicl spce is identicl to the originl M. Lemm Let (M, g 1 ), (M, g 2 ) be Riemnnin mnifolds, nd suppose M is compct. Then there exist constnt λ > 0 such tht for ny p, q M we hve d g 1(p, q) λ d g 2(p, q). Proof. Define f : S n 1 M R by f(u, p) := g 1 p(u, u)/g 2 p(u, u). Note tht, since g is positive definite, f > 0. Thus since f is continuous nd S n 1 M is compct f λ 2 > 0. Consequently, bilinerity of g yields tht g 1 p(v, v) λ 2 g 2 p(v, v), for ll v R n. Next note tht the bove inequlity yields Length g1 [c] = g 1 c(t) (c (t), c (t)) dt λ g 2 c(t) (c (t), c (t)) dt = λ Length g2 [c]. for ny curve c: [, b] M. In prticulr the bove inequlities hold for ll curves c: [, b] M with c() = p nd c(b) = q. Proposition The metric spce (M, d g ), endowed with its metric topology, is homeomorphic to M with its stndrd topology. 7
8 Proof. There re two prts to this rgument: Prt I: We hve to show tht every open neighborhood U of M is open in its metric topology, i.e., for every p U there exists n r > 0 such tht B g r (p) U, where B g r (p) := {q M d g (p, q) < r}. To see this first note tht, s we showed in the proof of the previous proposition, there exists n open neighborhood V of p with V U such tht there exists homeomorphism φ: V B n 1 (o). Now, much s in the proof of the previous proposition, if we endow B n 1 (o) with the push forwrd metric induced by φ then (B n 1 (o), dφ(g)) becomes isometric to (V, g). But recll tht, s we showed in the erlier proposition, the distnce of ny point in the boundry B n 1 (o) = Sn of B n 1 (o) from the origin o ws bigger thn some constnt, sy λ. Thus the sme is true of the distnce of V from p. In prticulr, if we choose r < λ, then B g r (p) V U. Prt II: We hve to show tht every metric bll B g r (p) is open in M, i.e., t every q B g r (p) we cn find open neighborhood U of q in M such tht U B g r (p). To see this let V be n open neighborhood of p such tht there exists homeomorphism ψ : V B n 1 (o), nd endow B n 1 (o) with the push forwrd metric dψ(g). Then the distnce of ψ(v B g r (p)) from o is equl to r, with respect to the metric dψ(g). So, by the previous proposition, this distnce, with respect to the Eucliden metric on B n 1 (o) must be t lest λr > 0. Thus if we choose r < λ r, then the Eucliden bll B n r (o) ψ(v ). Consequently, U := ψ 1 (B n r (o)) V, nd U is open in M, since B n r (o) is open. 8
Notes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationMath Advanced Calculus II
Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More information440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
More informationII. Integration and Cauchy s Theorem
MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationNote 16. Stokes theorem Differential Geometry, 2005
Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationINDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012
Lecture 6: Line Integrls INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Anlysis Autumn 2012 August 8, 2012 Lecture 6: Line Integrls Lecture 6: Line Integrls Lecture 6: Line Integrls Integrls of complex
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationComplex integration. L3: Cauchy s Theory.
MM Vercelli. L3: Cuchy s Theory. Contents: Complex integrtion. The Cuchy s integrls theorems. Singulrities. The residue theorem. Evlution of definite integrls. Appendix: Fundmentl theorem of lgebr. Discussions
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationCalculus of Variations: The Direct Approach
Clculus of Vritions: The Direct Approch Lecture by Andrejs Treibergs, Notes by Bryn Wilson June 7, 2010 The originl lecture slides re vilble online t: http://www.mth.uth.edu/~treiberg/directmethodslides.pdf
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More information1.3 The Lemma of DuBois-Reymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis - January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationFor a continuous function f : [a; b]! R we wish to define the Riemann integral
Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationClassification of Spherical Quadrilaterals
Clssifiction of Sphericl Qudrilterls Alexndre Eremenko, Andrei Gbrielov, Vitly Trsov November 28, 2014 R 01 S 11 U 11 V 11 W 11 1 R 11 S 11 U 11 V 11 W 11 2 A sphericl polygon is surfce homeomorphic to
More informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More informationREGULARITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2
EGULAITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2 OVIDIU SAVIN AND ENICO VALDINOCI Abstrct. We show tht the only nonlocl s-miniml cones in 2 re the trivil ones for ll s 0, 1). As consequence we obtin tht
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently
More informationHYPERBOLIC PLANE AS A PATH METRIC SPACE. Contents
HYPERBOLIC PLANE AS A PATH METRIC SPACE QINGCI AN Abstrct. We will study the quntities tht re invrint under the ction of Möb(H) on hyperbolic plne, nmely, the length of pths in hyperbolic plne. From these
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationHandout 4. Inverse and Implicit Function Theorems.
8.95 Hndout 4. Inverse nd Implicit Function Theorems. Theorem (Inverse Function Theorem). Suppose U R n is open, f : U R n is C, x U nd df x is invertible. Then there exists neighborhood V of x in U nd
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationMAS 4156 Lecture Notes Differential Forms
MAS 4156 Lecture Notes Differentil Forms Definitions Differentil forms re objects tht re defined on mnifolds. For this clss, the only mnifold we will put forms on is R 3. The full definition is: Definition:
More informationThe Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f
More informationMatFys. Week 2, Nov , 2005, revised Nov. 23
MtFys Week 2, Nov. 21-27, 2005, revised Nov. 23 Lectures This week s lectures will be bsed on Ch.3 of the text book, VIA. Mondy Nov. 21 The fundmentls of the clculus of vritions in Eucliden spce nd its
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationOne dimensional integrals in several variables
Chpter 9 One dimensionl integrls in severl vribles 9.1 Differentition under the integrl Note: less thn 1 lecture Let f (x,y be function of two vribles nd define g(y : b f (x,y dx Suppose tht f is differentible
More informationRegulated functions and the regulated integral
Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationRudin s Principles of Mathematical Analysis: Solutions to Selected Exercises. Sam Blinstein UCLA Department of Mathematics
Rudin s Principles of Mthemticl Anlysis: Solutions to Selected Exercises Sm Blinstein UCLA Deprtment of Mthemtics Mrch 29, 2008 Contents Chpter : The Rel nd Complex Number Systems 2 Chpter 2: Bsic Topology
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationCHAPTER 4 MULTIPLE INTEGRALS
CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationDifferential Geometry: Conformal Maps
Differentil Geometry: Conforml Mps Liner Trnsformtions Definition: We sy tht liner trnsformtion M:R n R n preserves ngles if M(v) 0 for ll v 0 nd: Mv, Mw v, w Mv Mw v w for ll v nd w in R n. Liner Trnsformtions
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationc n φ n (x), 0 < x < L, (1) n=1
SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry
More informationSurface Integrals of Vector Fields
Mth 32B iscussion ession Week 7 Notes Februry 21 nd 23, 2017 In lst week s notes we introduced surfce integrls, integrting sclr-vlued functions over prmetrized surfces. As with our previous integrls, we
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationIndefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More information1. Review. t 2. t 1. v w = vw cos( ) where is the angle between v and w. The above leads to the Schwarz inequality: v w vw.
1. Review 1.1. The Geometry of Curves. AprmetriccurveinR 3 is mp R R 3 t (t) = (x(t),y(t),z(t)). We sy tht is di erentile if x, y, z re di erentile. We sy tht it is C 1 if, in ddition, the derivtives re
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationThe Form of Hanging Slinky
Bulletin of Aichi Univ. of Eduction, 66Nturl Sciences, pp. - 6, Mrch, 07 The Form of Hnging Slinky Kenzi ODANI Deprtment of Mthemtics Eduction, Aichi University of Eduction, Kriy 448-854, Jpn Introduction
More informationPart I. Some reminders. Definition (Euclidean n-space) Theorem (Cauchy s inequality) Definition (Open ball) Definition (Cross product)
Prt I ome reminders 1.1. Eucliden spces 1.2. ubsets of Eucliden spce 1.3. Limits nd continuity Definition (Eucliden n-spce) The set of ll ordered n-tuples of rel numbers, x = (x1,..., xn), is the Eucliden
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More information