Practice Integration Math 120 Calculus I D Joyce, Fall 2013

Size: px
Start display at page:

Download "Practice Integration Math 120 Calculus I D Joyce, Fall 2013"

Transcription

1 Pactice Integation Math 0 Calculus I D Joyce, Fall 0 This fist set of indefinite integals, that is, antideivatives, only depends on a few pinciples of integation, the fist being that integation is invese to diffeentiation. Besides that, a few ules can be identified: a constant ule, a powe ule, lineaity, and a limited few ules fo tigonometic, logaithmic, and eponential functions. k d = k + C, whee k is a constant n d = n + n+ + C, if n d = ln + C kf( d = k f( d (f( ± g( d = f( d ± sin d = cos + C cos d = sin + C e d = e + C d = actan + C + d = acsin + C g( d We ll add moe ules late, but thee ae plenty hee to get acquainted with. Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps ( + d. Hint. ( 8 t + t + dt. Hint. (u / + u / du. Hint. ( d. Hint. d. Hint. ( t + dt. Hint. t ( 5 y y dy. Hint. + + d. Hint. ( sin θ + cos θ dθ. Hint. (5e e d. Hint. dt. Hint. + t (e + + e d. Hint. du. Hint. u

2 ( + d. Hint. sin d. Hint. tan ( cos + e d. Hint. v dv. Hint. dt. Hint. d. Hint e d. Hint. Integating polynomials is faily easy, and you ll get the hang of it afte doing just a couple of them.. Hint. (u / + u / du. You can use the powe ule fo othe powes besides integes. Fo instance, u / du = 5 u5/ + C. Hint. ( d You can even use the powe ule fo negative eponents (ecept. Fo eample, d = + C. Hint. ( + d. Integate each tem using the powe ule, n d = n + n+ + C. So to integate n, incease the powe by, then divide by the new powe.. Hint. ( 8 t + t + dt. Remembe that the integal of a constant is the constant times the integal. Anothe way to say that is that you can pass a constant though the integal sign. Fo instance, 8 dt = 5 t 8 dt 5. Hint. d This is and the geneal powe ule doesn t apply. But you can use d = ln + C. 6. Hint. ( t + dt t Teat the fist tem as t and the second tem as t.. Hint. ( 5 y y dy It s usually easie to tun those squae oots into factional powes. So, fo instance, is y /. y

3 Hint. d Use some algeba to simplify the integand, that is, divide by befoe integating. 9. Hint. ( sin θ + cos θ dθ Getting the ± signs ight when integating sines and cosines takes pactice. 0. Hint. (5e e d Just as the deivative of e is e, so the integal of e is e. Note that the e in the integand is a constant.. Hint. + t dt Remembe that the deivative of actan t is + t.. Hint. (e + + e d When woking with eponential functions, emembe to use the vaious ules of eponentiation. Hee, the ules to use ae e a+b = e a e b and e a b = e a /e b.. Hint. u du 6. Hint. ( cos + e d Just moe pactice with tig and eponential functions.. Hint. v dv You can wite v as v. And emembe you can wite v as v /. 8. Hint. dt Use algeba to wite this in a fom that s easie to integate. Remembe that / t is t /. 9. Hint. + d You can facto out a fom the denominato to put it in a fom you can integate. 6 + e 0. Hint. d Divide though by befoe integating. Altenatively, wite the integand as and multiply. / ( 6 + e / Remembe that the deivative of acsin u is u. ( + d. (. Hint. + d Use the powe ule, but don t foget the integal of / is ln + C. sin 5. Hint. tan d You ll need to use tig identities to simplify this. The integal is C. Wheneve you e woking with indefinite integals like this, be sue to wite the +C. It signifies that you can add any constant to the antideivative F ( to get anothe one, F ( + C. When you e woking with definite integals with limits of integation, b a, the constant isn t needed since you ll be evaluating an antideivative F ( at b and a to get a numeical answe F (b F (a.

4 . ( 8 t + t + dt. The integal is 5 9 t9 5 t5 + t + t + C.. (u / + u / du. This integal evaluates as 5 u5/ + u/ + C.. ( d. That equals + +C. If you pefe, you could wite the answe as + + C 5. d That s ln +C. The eason the absolute value sign is thee is that when is negative, the deivative of ln is /, so by putting in the absolute value sign, you e coveing that case, too. 6. ( t + dt. t The integal of t + t is t + ln t +C.. ( 5 y y dy. The integal of 5y / y / is 0 y/ 6y / +C. You could wite that as 0 y y 6 y + C if you pefe d. The integal of + + is ln + C. ( sin θ + cos θ dθ. That s equal to cos θ + sin θ + C. 0. (5e e d That equals 5e e + C.. + t dt. That evaluates as actan t + C. pefe to wite actan t as tan t.. (e + + e d. Some people The integand is its own antideivative, that is, the integal is equal to e + + e + C. If you wite the integand as e e + e /e, and note that e is just a constant, you can see that it s its own antideivative.. u du. The integal equals acsin u.. ( + d. The integal evaluates as + ln + C. sin 5. tan d The integand simplifies to cos. Theefoe the integal is sin + C. 6. That s sin + e + C.. ( cos + e d. v dv. Since you can ewite the integand as v /, theefoe its integal is v / + C.

5 8. dt. The integal of t / is equal to 8 t / + C. 5 5 You could also wite that as 8 t/5 + C d This integal equals actan + C e d. The integal can be ewitten as ( / 6 5/ + e d which equals 9 9/ / + e + C. Math 0 Home Page at 5

Practice Integration Math 120 Calculus I Fall 2015

Practice Integration Math 120 Calculus I Fall 2015 Pactice Integation Math 0 Calculus I Fall 05 Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps... ( + d. Hint. Answe. ( 8 t + t + This fist set of indefinite

More information

radians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side

radians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

P.7 Trigonometry. What s round and can cause major headaches? The Unit Circle.

P.7 Trigonometry. What s round and can cause major headaches? The Unit Circle. P.7 Tigonomet What s ound and can cause majo headaches? The Unit Cicle. The Unit Cicle will onl cause ou headaches if ou don t know it. Using the Unit Cicle in Calculus is equivalent to using ou multiplication

More information

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!! Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time

More information

Chapter Eight Notes N P U1C8S4-6

Chapter Eight Notes N P U1C8S4-6 Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that

More information

What to Expect on the Placement Exam

What to Expect on the Placement Exam What to Epect on the Placement Eam Placement into: MTH o MTH 44 05 05 The ACCUPLACER placement eam is an adaptive test ceated by the College Boad Educational Testing Sevice. This document was ceated to

More information

MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates

MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v

More information

Markscheme May 2017 Calculus Higher level Paper 3

Markscheme May 2017 Calculus Higher level Paper 3 M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted

More information

Double-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities

Double-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities Double-angle & powe-eduction identities Pat 5, Tigonomety Lectue 5a, Double Angle and Powe Reduction Fomulas In the pevious pesentation we developed fomulas fo cos( β) and sin( β) These fomulas lead natually

More information

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!

More information

Tutorial Exercises: Central Forces

Tutorial Exercises: Central Forces Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total

More information

10.2 Parametric Calculus

10.2 Parametric Calculus 10. Paametic Calculus Let s now tun ou attention to figuing out how to do all that good calculus stuff with a paametically defined function. As a woking eample, let s conside the cuve taced out by a point

More information

B da = 0. Q E da = ε. E da = E dv

B da = 0. Q E da = ε. E da = E dv lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the

More information

Graphs of Sine and Cosine Functions

Graphs of Sine and Cosine Functions Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the

More information

Question 1: The dipole

Question 1: The dipole Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite

More information

Related Rates - the Basics

Related Rates - the Basics Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the

More information

Ch 8 Alg 2 Note Sheet Key

Ch 8 Alg 2 Note Sheet Key Ch 8 Alg Note Sheet Key Chapte 8: Eponential and Logaithmic Functions 8. Eploing Eponential Models Fo some data, the est model is a function that uses the independent vaiale as an eponent. An eponential

More information

15 Solving the Laplace equation by Fourier method

15 Solving the Laplace equation by Fourier method 5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the

More information

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal

More information

Physics 121 Hour Exam #5 Solution

Physics 121 Hour Exam #5 Solution Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given

More information

5.8 Trigonometric Equations

5.8 Trigonometric Equations 5.8 Tigonometic Equations To calculate the angle at which a cuved section of highwa should be banked, an enginee uses the equation tan =, whee is the angle of the 224 000 bank and v is the speed limit

More information

Compactly Supported Radial Basis Functions

Compactly Supported Radial Basis Functions Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically

More information

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following:

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following: Revie of the H-O model Poblem 1 Assume that the poduction functions in the standad H-O model ae the folloing: f 1 L 1 1 ) L 1/ 1 1/ 1 f L ) L 1/3 /3 In addition e assume that the consume pefeences ae given

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Chapter 2: Basic Physics and Math Supplements

Chapter 2: Basic Physics and Math Supplements Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate

More information

When a mass moves because of a force, we can define several types of problem.

When a mass moves because of a force, we can define several types of problem. Mechanics Lectue 4 3D Foces, gadient opeato, momentum 3D Foces When a mass moves because of a foce, we can define seveal types of poblem. ) When we know the foce F as a function of time t, F=F(t). ) When

More information

Name Date. Trigonometric Functions of Any Angle For use with Exploration 5.3

Name Date. Trigonometric Functions of Any Angle For use with Exploration 5.3 5.3 Tigonometic Functions of An Angle Fo use with Eploation 5.3 Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with,

More information

PHYS 301 HOMEWORK #10 (Optional HW)

PHYS 301 HOMEWORK #10 (Optional HW) PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2

More information

1 Similarity Analysis

1 Similarity Analysis ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial

More information

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic. Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus

More information

PDF Created with deskpdf PDF Writer - Trial ::

PDF Created with deskpdf PDF Writer - Trial :: A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees

More information

MATH section 2.7 Related Rates Page 1 of 7

MATH section 2.7 Related Rates Page 1 of 7 MATH 0100 section.7 Related Rates Page 1 of 7 Unfotunatel, thee isn t much I can infom befoe ou encounte difficulties in this section. Remembe that this section is all wod poblems. You must be able to

More information

Mark Scheme 4727 June 2006

Mark Scheme 4727 June 2006 Mak Scheme 77 June 006 77 Mak Scheme June 006 (a) Identity = + 0 i Invese = + i i = + i i 0 0 (b) Identity = 0 0 0 Invese = 0 0 i B Fo coect identity. Allow B Fo seen o implied + i = B Fo coect invese

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80

0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS Intenational Geneal Cetificate of Seconday Education MARK SCHEME fo the Octobe/Novembe 009 question pape fo the guidance of teaches 0606 ADDITIONAL MATHEMATICS

More information

Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf

Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf Econ 0A Poblem Set 4 Solutions ue in class on Tu 4 Novembe. No late Poblem Sets accepted, so! This Poblem set tests the knoledge that ou accumulated mainl in lectues 5 to 9. Some of the mateial ill onl

More information

Physics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving

Physics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act,

More information

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006 1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and

More information

So, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur =

So, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur = 3.4 Geen s Theoem Geoge Geen: self-taught English scientist, 793-84 So, if we ae finding the amount of wok done ove a non-consevative vecto field F, we do that long u b u 3. method Wok = F d F( () t )

More information

Class #16 Monday, March 20, 2017

Class #16 Monday, March 20, 2017 D. Pogo Class #16 Monday, Mach 0, 017 D Non-Catesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y =

More information

Single Particle State AB AB

Single Particle State AB AB LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.

More information

Physics 521. Math Review SCIENTIFIC NOTATION SIGNIFICANT FIGURES. Rules for Significant Figures

Physics 521. Math Review SCIENTIFIC NOTATION SIGNIFICANT FIGURES. Rules for Significant Figures Physics 51 Math Review SCIENIFIC NOAION Scientific Notation is based on exponential notation (whee decimal places ae expessed as a powe of 10). he numeical pat of the measuement is expessed as a numbe

More information

REVIEW Polar Coordinates and Equations

REVIEW Polar Coordinates and Equations REVIEW 9.1-9.4 Pola Coodinates and Equations You ae familia with plotting with a ectangula coodinate system. We ae going to look at a new coodinate system called the pola coodinate system. The cente of

More information

, the tangent line is an approximation of the curve (and easier to deal with than the curve).

, the tangent line is an approximation of the curve (and easier to deal with than the curve). 114 Tangent Planes and Linea Appoimations Back in-dimensions, what was the equation of the tangent line of f ( ) at point (, ) f ( )? (, ) ( )( ) = f Linea Appoimation (Tangent Line Appoimation) of f at

More information

Homework # 3 Solution Key

Homework # 3 Solution Key PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in

More information

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r. The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the

More information

Figure 1. We will begin by deriving a very general expression before returning to Equations 1 and 2 to determine the specifics.

Figure 1. We will begin by deriving a very general expression before returning to Equations 1 and 2 to determine the specifics. Deivation of the Laplacian in Spheical Coodinates fom Fist Pinciples. Fist, let me state that the inspiation to do this came fom David Giffiths Intodction to Electodynamics textbook Chapte 1, Section 4.

More information

Math 124B February 02, 2012

Math 124B February 02, 2012 Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial

More information

Math Section 4.2 Radians, Arc Length, and Area of a Sector

Math Section 4.2 Radians, Arc Length, and Area of a Sector Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic

More information

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating

More information

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3. Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.

More information

Berkeley Math Circle AIME Preparation March 5, 2013

Berkeley Math Circle AIME Preparation March 5, 2013 Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.

More information

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,

More information

A New Approach to General Relativity

A New Approach to General Relativity Apeion, Vol. 14, No. 3, July 7 7 A New Appoach to Geneal Relativity Ali Rıza Şahin Gaziosmanpaşa, Istanbul Tukey E-mail: aizasahin@gmail.com Hee we pesent a new point of view fo geneal elativity and/o

More information

Part V: Closed-form solutions to Loop Closure Equations

Part V: Closed-form solutions to Loop Closure Equations Pat V: Closed-fom solutions to Loop Closue Equations This section will eview the closed-fom solutions techniques fo loop closue equations. The following thee cases will be consideed. ) Two unknown angles

More information

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009 Physics 111 Lectue 5 (Walke: 3.3-6) Vectos & Vecto Math Motion Vectos Sept. 11, 2009 Quiz Monday - Chap. 2 1 Resolving a vecto into x-component & y- component: Pola Coodinates Catesian Coodinates x y =

More information

A Hartree-Fock Example Using Helium

A Hartree-Fock Example Using Helium Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty June 6 A Hatee-Fock Example Using Helium Cal W. David Univesity of Connecticut, Cal.David@uconn.edu Follow

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS 6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to

More information

Numerical Integration

Numerical Integration MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.

More information

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,

More information

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0

More information

Rigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018

Rigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018 Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining

More information

8 Separation of Variables in Other Coordinate Systems

8 Separation of Variables in Other Coordinate Systems 8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies

More information

ADVANCED SUBSIDIARY (AS) General Certificate of Education Mathematics Assessment Unit F1. assessing. Module FP1: Further Pure Mathematics 1

ADVANCED SUBSIDIARY (AS) General Certificate of Education Mathematics Assessment Unit F1. assessing. Module FP1: Further Pure Mathematics 1 ADVACED SUBSIDIARY (AS) Geneal Cetificate of Education 15 Mathematics Assessment Unit F1 assessing Module F1: Futhe ue Mathematics 1 [AMF11] WEDESDAY 4 UE, MRIG MAR SCHEME 958.1 F GCE ADVACED/ADVACED SUBSIDIARY

More information

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2. Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed

More information

0606 ADDITIONAL MATHEMATICS

0606 ADDITIONAL MATHEMATICS UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS Intenational Geneal Cetificate of Seconday Education MARK SCHEME fo the Octobe/Novembe 011 question pape fo the guidance of teaches 0606 ADDITIONAL MATHEMATICS

More information

x x2 2 B A ) v(0, t) = 0 and v(l, t) = 0. L 2. This is a familiar heat equation initial/boundary-value problem and has solution

x x2 2 B A ) v(0, t) = 0 and v(l, t) = 0. L 2. This is a familiar heat equation initial/boundary-value problem and has solution Hints to homewok 7 8.2.d. The poblem is u t ku xx + k ux fx u t A u t B. It has a souce tem and inhomogeneous bounday conditions but none of them depend on t. So as in example 3 of the notes we should

More information

KEPLER S LAWS OF PLANETARY MOTION

KEPLER S LAWS OF PLANETARY MOTION EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee

More information

Mechanics and Special Relativity (MAPH10030) Assignment 3

Mechanics and Special Relativity (MAPH10030) Assignment 3 (MAPH0030) Assignment 3 Issue Date: 03 Mach 00 Due Date: 4 Mach 00 In question 4 a numeical answe is equied with pecision to thee significant figues Maks will be deducted fo moe o less pecision You may

More information

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit

More information

The Substring Search Problem

The Substring Search Problem The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is

More information

Calculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008

Calculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008 Calculus I Section 4.7 Optimization Solutions Math 151 Novembe 9, 008 The following poblems ae maimum/minimum optimization poblems. They illustate one of the most impotant applications of the fist deivative.

More information

Chapter 5: Integrals

Chapter 5: Integrals Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental

More information

Math 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3

Math 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3 Math : alculus I Math/Sci majos MWF am / pm, ampion Witten homewok Review: p 94, p 977,8,9,6, 6: p 46, 6: p 4964b,c,69, 6: p 47,6 p 94, Evaluate the following it by identifying the integal that it epesents:

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Kepler s problem gravitational attraction

Kepler s problem gravitational attraction Kele s oblem gavitational attaction Summay of fomulas deived fo two-body motion Let the two masses be m and m. The total mass is M = m + m, the educed mass is µ = m m /(m + m ). The gavitational otential

More information

AP Physics Electric Potential Energy

AP Physics Electric Potential Energy AP Physics lectic Potential negy Review of some vital peviously coveed mateial. The impotance of the ealie concepts will be made clea as we poceed. Wok takes place when a foce acts ove a distance. W F

More information

Electric Charge and Field

Electric Charge and Field lectic Chage and ield Chapte 6 (Giancoli) All sections ecept 6.0 (Gauss s law) Compaison between the lectic and the Gavitational foces Both have long ange, The electic chage of an object plas the same

More information

MAC Module 12 Eigenvalues and Eigenvectors

MAC Module 12 Eigenvalues and Eigenvectors MAC 23 Module 2 Eigenvalues and Eigenvectos Leaning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue poblem by finding the eigenvalues and the coesponding eigenvectos

More information

Partition Functions. Chris Clark July 18, 2006

Partition Functions. Chris Clark July 18, 2006 Patition Functions Chis Clak July 18, 2006 1 Intoduction Patition functions ae useful because it is easy to deive expectation values of paametes of the system fom them. Below is a list of the mao examples.

More information

Classical Mechanics Homework set 7, due Nov 8th: Solutions

Classical Mechanics Homework set 7, due Nov 8th: Solutions Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with

More information

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface . CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),

More information

Radian and Degree Measure

Radian and Degree Measure CHAT Pe-Calculus Radian and Degee Measue *Tigonomety comes fom the Geek wod meaning measuement of tiangles. It pimaily dealt with angles and tiangles as it petained to navigation, astonomy, and suveying.

More information

Algebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9.

Algebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9. lgeba Substitution in algeba Remembe... In an algebaic expession, lettes ae used as substitutes fo numbes. Example Find the value of the following expessions if s =. a) s + + = = s + + = = Example Find

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

CHAPTER 3. Section 1. Modeling Population Growth

CHAPTER 3. Section 1. Modeling Population Growth CHAPTER 3 Section 1. Modeling Population Gowth 1.1. The equation of the Malthusian model is Pt) = Ce t. Apply the initial condition P) = 1. Then 1 = Ce,oC = 1. Next apply the condition P1) = 3. Then 3

More information

Introduction to Arrays

Introduction to Arrays Intoduction to Aays Page 1 Intoduction to Aays The antennas we have studied so fa have vey low diectivity / gain. While this is good fo boadcast applications (whee we want unifom coveage), thee ae cases

More information

Math Notes on Kepler s first law 1. r(t) kp(t)

Math Notes on Kepler s first law 1. r(t) kp(t) Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ... MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto

More information

Curvature singularity

Curvature singularity Cuvatue singulaity We wish to show that thee is a cuvatue singulaity at 0 of the Schwazschild solution. We cannot use eithe of the invaiantsr o R ab R ab since both the Ricci tenso and the Ricci scala

More information

Handout: IS/LM Model

Handout: IS/LM Model Econ 32 - IS/L odel Notes Handout: IS/L odel IS Cuve Deivation Figue 4-4 in the textbook explains one deivation of the IS cuve. This deivation uses the Induced Savings Function fom Chapte 3. Hee, I descibe

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

DonnishJournals

DonnishJournals DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş

More information

dq 1 (5) q 1 where the previously mentioned limit has been taken.

dq 1 (5) q 1 where the previously mentioned limit has been taken. 1 Vecto Calculus And Continuum Consevation Equations In Cuvilinea Othogonal Coodinates Robet Maska: Novembe 25, 2008 In ode to ewite the consevation equations(continuit, momentum, eneg) to some cuvilinea

More information

10/04/18. P [P(x)] 1 negl(n).

10/04/18. P [P(x)] 1 negl(n). Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the

More information