# 9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes

Size: px
Start display at page:

Download "9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes"

Transcription

1 The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen ecuse of its mny pplictions. When vectors re multiplied using the vector product the result is lwys vector. Prerequisites Before strting this Section you should... Lerning Outcomes After completing this Section you should e le to... 1 tht vector cn e represented s directed line segment 2 how to express vector in crtesin form 3 how to evlute 3 3 determinnts understnd the right-hnded screw rule clculte the vector product of two given vectors use determinnts to clculte the vector product of two vectors given in crtesin form

2 1. The Right-hnded Screw Rule To understnd how the vector product is formed it is helpful to consider first the right-hnded screw rule. Consider the two vectors nd shown in Figure 1. q Figure 1. The two vectors lie in plne; this plne is shded in Figure 1. Figure 2 shows the sme two vectors nd the plne in which they lie together with unit vector, denoted ê, which is perpendiculr to this plne. Imgine turning right-hnded screw, ligned long ê, inthe sense from towrds s shown. A right-hnded screw is one which when turned clockwise enters the mteril into which it is eing screwed (most screws re of this kind). You will see from Figure 2 tht the screw will dvnce in the direction of ê. ê Figure 2. On the other hnd, if the right-hnded screw is turned from towrds the screw will retrct in the direction of ˆf s shown in Figure 3. ˆf Figure 3. We re now in position to descrie the vector product. HELM (VERSION 1: Mrch 18, 2004): Workook Level 1 2

3 2. Definition of the Vector Product We define the vector product of nd, written s = sin ê By inspection of this formul note tht this is vector of mgnitude sin in the direction of the vector ê, where ê is unit vector perpendiculr to the plne contining nd in sense defined y the right-hnded screw rule. The quntity is red s cross nd is sometimes referred to s the cross product. See Figure 4. length sin Formlly we hve Figure 4. is perpendiculr to the plne contining nd. Key Point vector product: = sin ê modulus of vector product: = sin Note tht sin gives the vector product its modulus wheres ê gives its direction. Now study Figure 5 which is used to illustrte the clcultion of. In prticulr note the direction of rising through the ppliction of the right-hnded screw rule. We see tht is not equl to ecuse their directions re oppositely directed. In fct =. Exmple If nd re prllel, show tht =0. If nd re prllel then the ngle etween them is zero. Consequently sin =0from which it follows tht =0. Note tht the result, 0, isthe zero vector. Note in prticulr the following importnt results: 3 HELM (VERSION 1: Mrch 18, 2004): Workook Level 1

4 Figure 5. Clcultion of. Key Point i i =0 j j =0 k k =0 Exmple Show tht i j = k nd find expressions for j k nd k i. Note tht i nd j re perpendiculr so tht the ngle etween them is 90.Sothe modulus of i j is (1)(1) sin 90 =1. The unit vector perpendiculr to i nd j in the sense defined y the right-hnd screw rule is k s shown in the figure elow (left digrm). Therefore i j = k s required. z z z k i j =k j k =i k k k i =j x i j y j y i i x x The vector k is perpendiculr to oth i nd j. j y Similrly you should verify (see middle nd right-hnd digrm of the ove figure) tht j k = i nd k i = j. HELM (VERSION 1: Mrch 18, 2004): Workook Level 1 4

5 Key Point i j = k, j k = i, k i = j j i = k, k j = i, i k = j To help rememer these results you might like to think of the vectors i, j nd k written in lpheticl order like this: i j k i j k Moving left to right yields positive result: e.g. k i = j. Moving right to left yields negtive result s in j i = k 3. A Formul for Finding the Vector Product We cn use the oxed results of the previous section to develop formul for finding the vector product of two vectors given in crtesin form: Suppose = 1 i+ 2 j+ 3 k nd = 1 i+ 2 j+ 3 k then = ( 1 i + 2 j + 3 k) ( 1 i + 2 j + 3 k) = 1 i ( 1 i + 2 j + 3 k) + 2 j ( 1 i + 2 j + 3 k) + 3 k ( 1 i + 2 j + 3 k) = 1 1 (i i)+ 1 2 (i j)+ 1 3 (i k) (j i)+ 2 2 (j j)+ 2 3 (j k) (k i)+ 3 2 (k j)+ 3 3 (k k) Using the Key Point results, on pge 5, (ove) this expression simplifies to =( )i ( )j +( )k If = 1 i + 2 j + 3 k nd = 1 i + 2 j + 3 k then Key Point =( )i ( )j +( )k 5 HELM (VERSION 1: Mrch 18, 2004): Workook Level 1

6 Exmple Evlute the vector product if =3i 2j +5k nd =7i +4j 8k. Identifying 1 =3, 2 = 2, 3 =5, 1 =7, 2 =4, 3 = 8 wefind = (( 2)( 8) (5)(4))i ((3)( 8) (5)(7))j + ((3)(4) ( 2)(7))k = 4i +59j +26k Use the Key Point formul directly ove to find the vector product of p =3i +5j nd q =2i j. Your solution Note tht in this exmple there re no k components so 3 nd 3 re oth zero. Apply the formul: p q = 13k 4. Using Determinnts to Evlute Vector Product Evlution of vector product using the previous formul is very cumersome. A more convenient nd esily rememered method is to use determinnts. Recll tht, for 3 3 determinnt, c d e f g h i = e f h i d f g i + c d e g h The vector product of two vectors = 1 i + 2 j + 3 k nd = 1 i + 2 j + 3 k cn e found y evluting the determinnt: i j k = in which i, j nd k re (temporrily) treted s if they were sclrs. HELM (VERSION 1: Mrch 18, 2004): Workook Level 1 6

7 Key Point If = 1 i + 2 j + 3 k nd = 1 i + 2 j + 3 k then i j k = = i( ) j( )+k( ) Exmple Find the vector product of =3i 4j +2k nd =9i 6j +2k. We hve i j k = Evluting this determinnt we otin = i( 8 ( 12)) j(6 18) + k( 18 ( 36)) = 4i +12j +18k Exmple The re of tringle The re A T of the tringle shown in the figure elow is given y the formul A T = 1 c sin α. Show tht n equivlent formul is 2 A T = 1 AB AC. 2 A α c B C From the definition of the vector product AB AC = AB AC sin α since α is the ngle etween AB nd AC. Furthermore AB = c nd AC =. The required result follows immeditely. 7 HELM (VERSION 1: Mrch 18, 2004): Workook Level 1

8 Moments The moment (or torque) of the force F out point O is defined s M o = r F where r is position vector from O to ny point on the line of ction of F. F O r D It my seem strnge tht ny point on the line of ction my e tken ut it is esy to show tht exctly the sme vector is otined. By the properties of the cross product the direction of M o is perpendiculr to the plne contining r nd F (i.e. out of the pper). The mgnitude of the moment is M 0 = r F sin From the digrm r sin = D. Hence M o = D F. This would e the sme no mtter which point on the line of ction of F ws chosen. Exmple Find the moment of the force given y F =3i +4j +5k (N) cting t the point (14, 3, 6) out the point P (2, 2, 1). z r P (2, 2, 1) F =3i +4j +5k (14, 3, 6) y x HELM (VERSION 1: Mrch 18, 2004): Workook Level 1 8

9 The position vector r cn e ny vector from the point P to ny point on the line of ction of F. We cn tke (in metres) r = (14 2)i +( 3 ( 2))j +(6 1)k =12i j +5k. The moment is i j k M = r F = = 25i 45j +51k (Nm) Exercises 1. Show tht if nd re prllel vectors then their vector product is the zero vector. 2. Find the vector product of p = 2i 3j nd q =4i +7j. 3. If = i +2j +3k nd =4i +3j +2k find. Show tht. 4. Points A, B nd C hve coordintes (9, 1, 2), (3,1,3), nd (1, 0, 1) respectively. Find the vector product AB AC. 5. Find vector which is perpendiculr to oth of the vectors = i +2j +7k nd = i + j 2k. Hence find unit vector which is perpendiculr to oth nd. 6. Find vector which is perpendiculr to the plne contining 6i + k nd 2i + j. 7. For the vectors =4i +2j + k, = i 2j + k, nd c =3i 3j +4k, evlute oth ( c) nd ( ) c. Deduce tht, in generl, the vector product is not ssocitive. 8. Find the re of the tringle with vertices t the points with coordintes (1, 2, 3), (4, 3, 2) nd (8, 1, 1). 9. For the vectors r = i +2j +3k, s =2i 2j 5k, nd t = i 3j k, evlute ) (r t)s (s t)r. ) (r s) t. Deduce tht (r t)s (s t)r =(r s) t. 1 Answers 2. 2k 3. 5i +10j 5k 4. 5i 34j +6k 5. 11i +9j k, 203 ( 11i +9j k) 6. i +2j +6k for exmple. 7. 7i 17j +6k, 42i 46j 3k i 10j + k 9 HELM (VERSION 1: Mrch 18, 2004): Workook Level 1

### 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

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

### S56 (5.3) Vectors.notebook January 29, 2016

Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

### 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

### On the diagram below the displacement is represented by the directed line segment OA.

Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

### VECTORS, TENSORS, AND MATRICES. 2 + Az. A vector A can be defined by its length A and the direction of a unit

GG33 Lecture 7 5/17/6 1 VECTORS, TENSORS, ND MTRICES I Min Topics C Vector length nd direction Vector Products Tensor nottion vs. mtrix nottion II Vector Products Vector length: x 2 + y 2 + z 2 vector

### set is not closed under matrix [ multiplication, ] and does not form a group.

Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

### 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

### Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

### Thomas Whitham Sixth Form

Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### 13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes

The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce

### Vectors. Introduction. Definition. The Two Components of a Vector. Vectors

Vectors Introduction This pper covers generl description of vectors first (s cn e found in mthemtics ooks) nd will stry into the more prcticl res of grphics nd nimtion. Anyone working in grphics sujects

### Coordinate geometry and vectors

MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

### Review of Gaussian Quadrature method

Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

### 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

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

### R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

### 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

### Parse trees, ambiguity, and Chomsky normal form

Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

### Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### 3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit

Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors.

### 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

### 1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

### A study of Pythagoras Theorem

CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl

### 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

### 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

### 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

### 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

### JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson

JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis

### 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

### Resistive Network Analysis

C H A P T E R 3 Resistive Network Anlysis his chpter will illustrte the fundmentl techniques for the nlysis of resistive circuits. The methods introduced re sed on the circuit lws presented in Chpter 2:

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### 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

### Problem Solving 7: Faraday s Law Solution

MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

### DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

### 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

### Computer Graphics (CS 4731) Lecture 7: Linear Algebra for Graphics (Points, Scalars, Vectors)

Computer Grphics (CS 4731) Lecture 7: Liner Alger for Grphics (Points, Sclrs, Vectors) Prof Emmnuel Agu Computer Science Dept. Worcester Poltechnic Institute (WPI) Annoncements Project 1 due net Tuesd,

### Stage 11 Prompt Sheet

Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

### LINEAR ALGEBRA APPLIED

5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

### 1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### 13.4 Work done by Constant Forces

13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

### 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

### 1B40 Practical Skills

B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

### Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

### Conducting Ellipsoid and Circular Disk

1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

### Line Integrals. Chapter Definition

hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It

### Numerical Linear Algebra Assignment 008

Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com

### Network Analysis and Synthesis. Chapter 5 Two port networks

Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

### Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

### 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

### 13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes

The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce

### An Overview of Integration

An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

### 1. A refinement of a classical class number relation We give a refinement, and a new proof, of the following classical result [1, 2, 3].

A COMBINATORIAL REFINEMENT OF THE KRONECKER-HURWITZ CLASS NUMBER RELATION rxiv:604.08v [mth.nt] Apr 06 ALEXANDRU A. POPA AND DON ZAGIER Astrct. We give refinement of the Kronecker-Hurwitz clss numer reltion,

### Lecture Solution of a System of Linear Equation

ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

### What else can you do?

Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

### 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 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

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

### 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

Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd

### September 13 Homework Solutions

College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

### MEP Practice Book ES19

19 Vectors M rctice ook S19 19.1 Vectors nd Sclrs 1. Which of the following re vectors nd which re sclrs? Speed ccelertion Mss Velocity (e) Weight (f) Time 2. Use the points in the grid elow to find the

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### 16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings

Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the

### 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

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

### What s in Chapter 13?

Are nd volume 13 Wht s in Chpter 13? 13 01 re 13 0 Are of circle 13 03 res of trpeziums, kites nd rhomuses 13 04 surfce re of rectngulr prism 13 05 surfce re of tringulr prism 13 06 surfce re of cylinder

### along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

### 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.

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### Torsion in Groups of Integral Triangles

Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

### Minimal DFA. minimal DFA for L starting from any other

Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

### 3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration

Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,

### 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

### KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

### 14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes

Lengths of curves nd surfces of revolution 4.4 Introduction Integrtion cn be used to find the length of curve nd the re of the surfce generted when curve is rotted round n xis. In this section we stte

### Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

### To Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y

Foundtions of Computer Grphics (Spring 2010) CS 184, Lecture 2: Review of Bsic Mth http://inst.eecs.erkeley.edu/~cs184 o Do Complete Assignment 0 Downlod nd compile skeleton for ssignment 1 Red instructions

### The Evaluation Theorem

These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not

### ( 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

### 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

### Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

### r 0 ( ) cos( ) r( )sin( ). 1. Last time, we calculated that for the cardioid r( ) =1+sin( ),

Wrm up Recll from lst time, given polr curve r = r( ),, dx dy dx = dy d = (r( )sin( )) d (r( ) cos( )) = r0 ( )sin( )+r( ) cos( ) r 0 ( ) cos( ) r( )sin( ).. Lst time, we clculted tht for crdioid r( )

### Math 259 Winter Solutions to Homework #9

Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

### Chapter 7: Applications of Integrals

Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As

### Hints for Exercise 1 on: Current and Resistance

Hints for Exercise 1 on: Current nd Resistnce Review the concepts of: electric current, conventionl current flow direction, current density, crrier drift velocity, crrier numer density, Ohm s lw, electric

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### Trigonometric Functions

Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds