Size: px
Start display at page:

Transcription

1 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 the ply ground for the mthemticl study of curves in the -dimensionl plne nd the 3-dimensionl spce. In prticulr, we will discuss pproprite mthemticl representtion of curves, dene their tngent vectors nd rc lengths, nd derive formuls for these quntities. The required textook sections re.,.. The optionl textook sections re.3,.4,.5. The exmples in this note re dierent from exmples in the textook. Plese red the textook crefully nd try your hnds on the exercises. During this plese don't hesitte to contct me if you hve ny questions. Tle of contents Lecture 3: Curves in Clculus Prmetriztion of Curves Mthemticl representtions of curve Tngent Vectors nd Arc Length From possile velocities to ngent vectors Arc length of prmetrized curve Arc Length Prmetriztion Properties of curves from its derivtives

2 Dierentil Geometry of Curves & Surfces. Prmetriztion of Curves The most convenient wy (for ppliction of clculus) to represent sptil curve mthemticlly is to tret s the trjectory of moving prticle. This leds to the so-clled prmetriztion of the curves. There re innitely mny possile prmetriztions of the sme curve... Mthemticl representtions of curve Level curves. Before the dvent of clculus, curve is usully dened through level sets: i. (in the plne) s level sets: f(x; y) = ; ii. (in the spce) s intersection of surfces (intersection of level sets): Exmple. A circle in R is represented s A stright line in R 3 is represented s f (x; y; z) = ; f (x; y; z) = : () f(x; y) = (x ) + (y ) r = : () x + x + 3 x 3 = ; x + x + 3 x 3 = : (3) The sic ide is to replce the study of the curve y the study of one or more simple surfces. Prmetrized curves. To pply clculus, the most convenient representtion of curve is through prmetriztion. Definition. (Definition.. in textook) A prmetrized curve in R n is mp : (; ) 7! R n, for some ; with 6 < 6. Note. Recll tht (; ) mens the set of ll numers > nd <, while [; ] mens the set of ll numers > nd 6. Exercise. Do you rememer/cn you guess wht (; ] nd [; ) mens? Exmple 3. A circle in R is represented s x(t) + r cos t (t) = = y(t) + r sin t for t (; ) with < ; >. (4). The reson for such trnsition is tht, without clculus, there is no wy to study curve directly. Insted one hs to look for simple functions tht could generte the curve nd turn the study of the curves to the study of those functions.. Another nme for function.

3 Mth 348 Fll 7 Exmple 4. A strightline in R 3 is represented s (t) x(t) x(t) x B y A+ x(t) z for t ( ; ). u v w C A (5) Exmple 5. (Cycloid) The trjectory of unit circle rolling in the plne long line. Figure. The cycloids We tke t = s the prmeter. We hve hgmmi(t) = (t sin t; cos t) for t [; ]. Exercise. Write down prmetrized representtion of unit circle rolling outside (epicycloid) or inside (hypocycloid) nother circle with rdius R centered t the origin (for the inside cse, ssume R > ). Plot the following two cses: Outside with R = (Crdioid); Inside with R = 4 (Astroid). 3 Wht hppens if R!? Exmple 6. (Vivini's curve) The intersection of sphere x + y + z = R with x + y = R x. It cn e prmetrized y (t) = (R cos t; R cos t sin t; R sin t); t [; ) (6) Exercise 3. Check Ex...8 in the textook for wht looks like. Compre (6) with the formul there. Cn you nd dierent prmetriztion? Exercise 4. Consider plne curve given y grph: y = f(x). Is it level curve or prmetrized curve? Exercise 5. Try to drw the prmetrized curve (t) = (cos t; sin t; t), t (; 3 ). Exercise 6. Try to drw (t) = (t 3 ; t ) for t ( ; ). Remrk 7. Is cler tht neither level set nor prmetrized representtions re unique. There re innitely mny pirs of plnes thntersect long the sme stright line; There re innitely mny (t): (; ) 7! R n such tht their imge set coincide. We emphsize tht this non-uniqueness is in fct good thing s it llows us to choose the most convenient of them. We will see how this works lter cos t R R cos t + t ; + sin t R R R R sin t + t. R R 3

4 Dierentil Geometry of Curves & Surfces In 348 we will only consider smooth curves, ths prmetrized curves (t) = (x (t); x (t); :::; x n (t)), t (; ), with ech x n (t) innitely dierentile for ll t (; ). Exercise 7. Show through exmples tht the imge of smooth curve my not look smooth.. Tngent Vectors nd Arc Length When curve is viewed s the trjectory of moving prticle, its velocities re represented mthemticlly s tngent vectors, nd the length of the trjectory is the rc length... From possile velocities to ngent vectors From possile velocities to tngent vectors. Let (t) R n e the trjectory of prticle. Then its velocity t t is given y (t) (t ) v(t ) = lim = d t!t t t (t ) = _(t ): (7) For prmetrized curve (t), we will view it s mthemticl model of trjectory of prticle nd dene its velocity t (t ) to e one tngent vector t (t ). Definition 8. (Tngent vector) Let C e curve in R n. Let x C. Then v R n is tngent vector of C t x if nd only if there is prmetriztion (t): (; ) 7! R n of C such tht i. there is t (; ) with (t ) = x ; ii. _(t ) = v. Exercise 8. Let C e curve in R n. Let x C. Let v e tngent vector of C t x. i. Prove tht v is lso tngent vector of C t x ; ii. Prove tht v is lso tngent vector of C t x ; iii. Convince yourself tht for ritrry c R, c v is tngent vector of C t x ; iv. (optionl) Prove tht for ritrry c R, c v is tngent vector of C t x. The tngent vector of (t) t x = (t ): _(t) Regulr curves. A prmetrized curve (t): (; ) 7! R n is regulr if nd only if is smooth nd _(t) =/ for ll t (; ). Exmple 9. The prmetrized curve (t) = check i. t; t ; t 3 re ll smooth; t t C A is regulr. To see this we t 3 4

5 Mth 348 Fll 7 ii. _(t) = B t 3 t A no mtter wht s. From now on curve mens regulr curve... Arc length of prmetrized curve 5 The rc length formul. Let (t): (; ) 7! R n e prmetrized curve. Let [; ] (; ). We try to otin its rc length L through the following: Pick = t < < t k =. Consider the stright line segments (t )(t ), (t )(t ), (t )(t 3 ), :::, (t k )(t k ). Then intuitively we hve L > the sum of the lengths of these line segments. Inspired y the ove, one denes L := sup =t <<t k = i= Xk k( ) ( )k (8) where we emphsize tht the supreme is tken over ll possile prtitions = t < < t k =. In prticulr, k N is not xed. The formul. Arc length from () to () = k_(t)k : Theorem. Let (t) e smooth prmetrized curve. Then its rc length from () to () is given y L = k_(t)k : (9) Proof. (optionl; You my wnt to red the ox fter the proof first) We prove in two steps.. L 6 R k_(t)k. By fundmentl theorem of clculus, we hve ( ) ( ) = _(t) : () Thus we hve Xk k( ) ( )k = X k t= t= k _(t) 6 X k_(t)k t= = k_(t)k : ()

6 Dierentil Geometry of Curves & Surfces. L > R k_(t)k. Let k N nd dene := + i ( ). Then we hve = t k < t < < t k =. By fundmentl theorem of clculus, we hve ( ) ( ) = _(t) = _( ) ( ) + R i () where " t R i := [_(t) _( )] = ti (s) ds # : (3) As x(t) is regulr curve, there is M > such tht Consequently k(s)k 6 M for ll s [; ]: (4) kr i k 6 M ds = M ( ) M ( ) = : (5) k Thus we hve Xk k( ) ( )k = X k i= i= k > X i= k > X i= k_( ) ( ) + R i k [k_( ) ( )k kr i k] k_( )k ( ) M ( ) k : (6) On the other hnd, we hve k_(t)k = [k_(t)k k_( )k] + +k_( )k ( ) 6 k_( )k ( ) + = k_( )k ( ) + t + k(s)kds 6 k_( )k ( ) + M ( ) = k_( )k ( ) + k_(t) _( )k M ( ) k : (7) 6

7 Mth 348 Fll 7 From this we conclude Xk i= k( ) ( )k s k!. This implies L := sup Xk =t <t <<t k = i= M ( ) k_(t)k >! (8) k k( ) ( )k > k_(t)k (9) nd ends the proof. Technicl Aside In the ove proof we hve used the following results from clculus. Supreme. Let A e collection (set) of numers. Then its supreme sup A is the smllest numer ths greter thn or equl to A. As consequence, to prove tht sup A 6, ll we need to do is to show tht for every A, there holds 6 ; to prove tht sup A >, ll we need to do is to nd one prticulr sequence of n A such tht lim n! ( n ) >. Exercise 9. Wht do we need to do to prove sup A > or sup A <? Tringle inequlity. Clssicl tringle inequlity: Vrint: kxk + kyk > kx + yk: () jkxk kykj 6 kx yk: () Generliztion: kx k + + kx k k > kx + + x k k: () Integrl: Together with the denite of Riemnn integrls, () yields the following inequlity for vector functions: kx(t)k > x(t) : (3) Exmples. Exmple. Clculte the circumference of the unit circle x + y =. 7

8 Dierentil Geometry of Curves & Surfces Solution. p Method. We clculte the curve length l of the grph y = x ; 6x6. Then the circumference is L = l. q l = + p x dx = p dx x / x=sin t = = = : (4) So the circumference is L=. / Method. We prmetrize x(t) = cos t; y(t) = sin t, 6 t <. Then L = p x (t) + y (t) Exmple. Clculte the rc length of the spce curve = : (5) for t from to. Solution. We hve L = x = cos t; y = sin t; z = t (6) = p x (t) + y (t) + z (t) p p = : (7) Exmple 3. Clculte the rc length of the cycloid (t sin t; cos t) from t = to. Solution. We hve p L = ( cos t) + (sin t) p = cos t r = sin t sin t = = 8: (8) Exmple 4. Clculte the rc length of the Limcon of Pscl (( + cos t) cos t; ( + cos t) sin t), t (; ). 8

9 Mth 348 Fll 7 Solution. We hve (t) = (cos t + cos t; sin t + sin t). p L = ( sin t sin t) + (cos t + cos t) p = cos t : (9) It turns out tht this integrl cnnot e clculted explicitly. Exercise. Clculte the rc length of x = cos 3 t; y = sin 3 t; t [; ). Exercise. (Optionl) Is cler tht rc length should e independent of prmetriztion. Prove this. 3. Arc Length Prmetriztion Among the innitely mny possile prmetriztions of the sme curve, one prticulr prmetriztion, clled rc length prmetriztion stnds out s the most convenient due to its ility to simplify clcultions. Motivtion. For generl prmetrized curve (t): (; ) 7! R n, in generl we hve k_(t)k vrying with t. As we will see lter, mny clcultions could e gretly simplied if k_(t)k = for ll t (; ). Such prmetriztion is clled the rc length prmetriztion of the curve. Exercise. In this cse we hve the rc length etween x(t ) nd x(t ) to e t t. Existence of rc length prmetriztion. Theorem 5. Let (t): (; ) 7! R n e regulr curve. Then there is strictly incresing function T : (; ) 7! (; ) such tht the curve (s) := (T (s)) is prmetrized y its rc length. Nottion. The convention is tht, when the prmetriztion is rc length prmetriztion, we use s s the vrile. Ths, when we write curve s x(s), we ssume is lredy prmetrized y rc length. Proof. (Optionl) See Proposition.3.6 in the textook. Exmples. Exmple 6. Consider the circle ( cos t; sin t) in the plne. We hve k_(t)k = k( sin t; cos t)k = : (3) Thus the rc length prmetriztion is ( cos(s/); sin(s/)). Exmple 7. Consider the spce curve x = cos t; y = sin t; z = t: (3) 9

10 Dierentil Geometry of Curves & Surfces The rc length prmetriztion is cos p p p s/ ; sin s/ ; s/. 4. Properties of curves from its derivtives Exmple 8. k(t)k = constnf nd only if _(t) (t) = for ll t. Proof. We hve The conclusion now follows. d k(t)k = d ((t) (t)) = _(t) (t): (3) Exmple 9. Let (t) e curve in R 3. Then the following two re equivlent. ) There is nonzero constnt vector such tht (t) = f(t) for some sclr function f(t) =/. ) _(t) (t) = for ll t. Proof. )=)). We hve _(t) (t) = f _ (t) f(t) = f _ (t) f(t) ( ) = : (33) )=)). Let (t) := (t) f(t) with f(t) := k(t)k. Then we hve = _(t) (t) = d (f(t) (t)) (f(t) (t)) = f _ (t) (t) + f(t) _ (t) (f(t) (t)) = f _ (t) (t) (f(t) (t)) + f(t) _ (t) (f(t) (t)) = f _ (t) f(t) ((t) (t)) + f (t) _ (t) (t) = f (t) _ (t) (t) : (34) As f(t) =/, we hve _ (t) (t) =, or _ (t) is prllel to (t). On the other hnd, s k(t)k = (t) = k(t)k =, y Exmple 8 there holds _ (t) (t) = or _ (t)?(t). k(t)k k(t)k Consequently _ (t) =, or (t) = is constnt vector. Exercise 3. Wht hppens if we drop the ssumption f(t) =/? Exmple. Let (t) e curve in R 3. Then the following re equivlent. ) There is nonzero constnt vector such tht?(t) for ll t; ) ((t) _(t)) (t) = for ll t.

11 Mth 348 Fll 7 Proof. )=)). We hve _(t) = d ( (t)) = ; (t) = d ( _(t)) = : (35) Thus if we set the 33 mtrix =( _ ), then =. Conseqeuently, reclling =/, ( _) = det = : (36) )=)). Cse. (t) _(t) = for ll t. Then y Exmple 9 (t) = f(t) for some constnt vector. Tke? nd the conclusion follows. Cse. (t) _(t) =/ for ny t. Let (t) := (t) _(t). As ( _) =, is contined in the plne spnned y nd _. Consequently is perpendiculr to the sme plne, nd Now let (t) := (t) k(t)k. Exercise 4. Prove tht (t) _ (t) =. (t) _(t) = ( _) ( ) = : (37) By similr rgument s in Exmple 9, we conclude tht _ (t) =, ths is constnt vector. Now the conclusion follows from (t) = for ll t.

Mth 348 Fll 017 Lecture 6: Isometry Disclimer. As we hve textook, this lecture note is for guidnce nd sulement only. It should not e relied on when rering for exms. In this lecture we nish the reliminry

### Section 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

### We 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

### UNIFORM 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,

### A 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

### The 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

### 10 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

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Space 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)

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Advanced 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

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

### Fundamental 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

### Review 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

### Week 10: Line Integrals

Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

### ( 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

### Lecture 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

### 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

### ODE: 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() =

### Line 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

### u(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

### Math 231E, Lecture 33. Parametric Calculus

Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### Farey 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

### Lecture 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

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

### Math 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

### Chapter 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

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Beginning Darboux Integration, Math 317, Intro to Analysis II

Beginning Droux Integrtion, Mth 317, Intro to Anlysis II Lets strt y rememering how to integrte function over n intervl. (you lerned this in Clculus I, ut mye it didn t stick.) This set of lecture notes

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

### SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus

SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is

Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

### ODE: 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 dierentil eqution (ODE) du f(t) dt with initil condition u() : Just

### Math 1B, lecture 4: Error bounds for numerical methods

Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

### 13.3 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

### Handout: Natural deduction for first order logic

MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes

### Math 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech

Mth 6455 Oct 10, 2006 1 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

### W. 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)

### We 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:

### Definite 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

### Improper 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

### 1 The Riemann Integral

The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

### INDIAN 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

### Math 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:

### Theoretical 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

### Bases for Vector Spaces

Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

### Partial 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

### Math 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

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Note 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

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### APPLICATIONS 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

### Definite 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

### Surface 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

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### Overview of Calculus I

Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

### Line integrals, and arc length of curves. Some intuitive explanations and definitions.

Line integrls, nd rc length of curves. Some intuitive explntions nd definitions. Version 2. 8//2006 Thnks Yhel for correcting some misprints. A new remrk hs lso een dded t the end of this document. Let

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

### 440-2 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

### PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS

PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?

### Advanced 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

### US01CMTH02 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

### 7.2 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

### Abstract 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

### Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

### Coalgebra, Lecture 15: Equations for Deterministic Automata

Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

### Section 17.2 Line Integrals

Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We

### Chapter 8.2: The Integral

Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### Curves. Differential Geometry Lia Vas

Differentil Geometry Li Vs Curves Differentil Geometry Introduction. Differentil geometry is mthemticl discipline tht uses methods of multivrible clculus nd liner lgebr to study problems in geometry. In

### Recitation 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

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Mathematics. Area under Curve.

Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### g 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

### DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.

398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts

### MAA 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

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

### 1. 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

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### STUDY 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

### MATH1050 Cauchy-Schwarz Inequality and Triangle Inequality

MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### f(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