# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

3 Pages

### l_notes02

Course: MATH 1802, Winter 2004
School: Macquarie
Rating:

Word Count: 574

#### Document Preview

II Vectors Lecture and Vector Algebra A set S of rays is called a direction if it satises the following laws: (1) Any two rays in S have the same direction. (2) Every ray that has the same direction as some member of S is in S . A vector A consists of a non-negative real number, called the magnitude of the vector, and a direction. We denote the magnitude of A by |A|. A vector A such that |A| = 1 is called a...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Australia >> Macquarie >> MATH 1802

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
II Vectors Lecture and Vector Algebra A set S of rays is called a direction if it satises the following laws: (1) Any two rays in S have the same direction. (2) Every ray that has the same direction as some member of S is in S . A vector A consists of a non-negative real number, called the magnitude of the vector, and a direction. We denote the magnitude of A by |A|. A vector A such that |A| = 1 is called a unit vector. We will now describe four basic algebraic operations with vectors in E3 : 1 Multiplication by a scalar Let A by a vector, and let c by a real number. Multpying A by the scalar c, we obtain a vector denoted by cA. The magnitude of the result is given by |cA| = |c||A|. The direction of cA is the same as the direction of A if c 0, and opposite to the direction of A if c < 0. 2 Addition of vectors The sum of two vectors A and B is denoted simply by A + B . We can dene this sum geometrically. Translate B such that its start point is the end-point of A. Then A + B will be the vector having the same start-point as A and the same end-point as B . 3 Scalar product (Dot product) The scalar product of two vectors A and B is a scalar quantity denoted by . A B . Its value is A . B = |A||B |cos, where is the angle made by A and B if we translate B such that it has the same start-point as A. We observe that |B |cos is the length of the projection of B on A. 1 B A A+B Figure 1: Vector addition The magnitude of A is equal to the square root of the dot product of A with itself: | A| = Hence A |A| = A A .A A .B is a unit vector with the same direction as A. 4 Vector product (Cross product) The cross product of two vectors A and B is a vector denoted by A B . The magnitude of the cross product is given by: |A B | = |A| |B |sin. Let A and B have the same start-point P and end-points Q1 and Q2 , respectively. Let M be the plane of A and B . The direction of A B is normal to M in the manner established by the right-hand rule: if a right hand is placed at P and the ngers are curling from P Q1 to T Q2 through the angle smaller than , then the thumb indicates the direction of A B . The following are some properties of these basic vector operations. 1. (a + b)C = aC + bC 2. (ab)C = a(bC ) 3. a(B + C ) = aB + aC 2 4. A + B = B + A 5. A . (B + C ) = A . B + A . C 6. A B = B A 3
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Macquarie - MATH - 1802
Lecture IIIVector Algebra in Cartesian CoordinatesLet us construct a Cartesian coordinates system in E3 . First we choose apoint O, called the origin. Then we chose three mutually perpendicular raysstarting from O. These rays are called the positive x
Macquarie - MATH - 1802
Lecture IV Analytic Geometry in E2 and E3First we review some basic facts of analytic geometry in E2 . Let us consider a Cartesian coordinate system in E2 . We denote by F [x, y ] an algebraic formula in the variables x and y . Any equation of the form F
Macquarie - MATH - 1802
Lecture VCalculus of OneVariable FunctionsLet us rst review some denitions in calculus on real numbers.In order to dene limit on the real numbers we will use the concept of funnelfunctions.Denition 1 A function (t) on [0, d] is called a funnel functi
Macquarie - MATH - 1802
Lecture VICalculus of Vector FunctionsdRdt2denotes the rst-order derivative of R(t), and that d tR ded2notes the second-order derivative of R(t). We introduce new notations for these2ijfunctions: dR = R(t) and d R = R(t). Let R(t) = a1 (t) + a2
Macquarie - MATH - 1802
Lecture VIIPaths and CurvesFirst we go through several basic notions about paths. Let R(t) on [a, b] be agiven path.Denition 1 R(t) is called elementary if for every pair (t1 , t2 ), with t1 and t2distinct in [a, b], R(t1 ) = R(t2 ).Denition 2 R(t)
Macquarie - MATH - 1802
Lecture VIIIScalar FieldsCylindrical Coordinates1Scalar FieldsDenition 1 Let D be a subset of E3 . A function f that associates each pointP in D to a real number f (P ) is called a scalar eld. D is called the domainof f .Denition 2 Let f be a scal
Macquarie - MATH - 1802
Lecture IXLinear Approximation1Onevariable FunctionsLet f be a onevariable function on a domain D. If f is dierentiable at x = c,we say f has a linear approximation, which we dene in the following way.Denition 1 If f is a function dierentiable at c,
Macquarie - MATH - 1802
Lecture XLinear ApproximationChain Rule1Linear Approximation; GradientWe say that a function has a linear approximation on a domain D if it has alinear approximation at any point P D.Theorem 1 If f has a linear approximation on a domain D, then f i
Macquarie - MATH - 1802
Lecture XIChain Rule: Elimination MethodLet w = f (x, y ) be a dierentiable function of x and y . The linear approximationof f is given byfapp = fx (x, y )x + fy (x, y )y.We introduce a new notation, the dierential notation for the increments f,x, y
Macquarie - MATH - 1802
Lecture XIITerminology for PointSets in Euclidean Spaces andMinimumMaximum TheoremsFirst let us take a short look at a problem that was on the exam. We aregiven a level curve (in E2 ) or a surface(in E3 ) and a point P on that curve orsurface. How to
Macquarie - MATH - 1802
Lecture XIIITwoVariable TestConstrained MaximumMinimum Problems1The twovariable testRecall that by the Critical Point Theorem, onlyif the gradient of a function fat P is 0 (i.e. P is a critical point for f ), can P be an extreme point of f .Let f (
Macquarie - MATH - 1802
Lecture XIVMultiple Integrals1Integrals of onevariable functionsFor a realvalued function f dened on an interval [a, b], the integral of f overb[a, b], denoted by a f dx, is dened as follows:nbf dx =alimn,max xi 0f (x )xi ,ii=1where a = x1
Macquarie - MATH - 1802
Lecture XVIterated IntegralsIn this lecture we look at methods to compute multiple integrals, introducingiterated integrals. First, let us dene the type of regions for which it can be used.These regions are called simple regions.Denition 1 A region R
Macquarie - MATH - 1802
Lecture XVIIntegrals in Polar, Cylindrical, or Spherical CoordinatesUsually, we write functions in the Cartesian coordinate system. Hence wewrite and compute multiple integrals in Cartesian coordinates. But there areother coordinate systems that can h
Macquarie - MATH - 1802
Lecture XVIICurvilinear Coordinates; Change of VariablesAs we saw in lecture 16, in E2 we can use the polar coordinates system.In this system, we have a xed point O and a xed ray Ox. The coordinatesof a point P are given by r, the distance from P to O
Macquarie - MATH - 1802
Lecture XVIIIChange of Variables; Vector Fields1Change of VariablesRecall from lecture 17 that we change variables in integrals by the followingformula:f (x(u, v ), y (u, v ) Rf (x, y )dxdy =Rxuyuxvyv dudv,where R and R are correspondin
Macquarie - MATH - 1802
Lecture XIXVisualizing Vector Fields; Line Integrals1Visualizing Vector FieldsRecall that a vector eld in E2 is a function of the formF (x, y ) = f1 (x, y ) + f2 (x, y )ij.We dene two concepts that help us visualize vector elds.Denition 1 Let C b
Macquarie - MATH - 1802
Lecture XXVector Line Integrals; Conservative Fields1Vector line integralsLet F be a vector eld of domain D. Let D be connected, i.e. for any two pointsP, P in D there is a curve C contained in D that goes from P to P . Let C bea nite directed curve
Macquarie - MATH - 1802
Lecture XXILine Integrals; Conservative Fields1Line integralsLet us recapitulate the basic notions reering to line integrals. For both scalarand vector elds, we can dene line integrals. For a scalar eld f on a curveC , the line integral is denoted b
Macquarie - MATH - 1802
Lecture XXIISurfacesRecall that an elementary region D contained in E2 is called convex if forany points in D, the line conecting them is contained in D. Also recall that amap R on D is called injective if distinct points in D give dierent values of R
Macquarie - MATH - 1802
Lecture XXIIISurface IntegralsRemember that the parametric expression for a curve is given by a vector function R(t) = x(t) + y (t) + z (t)k . The parametric expression for a surface Sijis given by a vector function of two variables: R(u, v ) = x(u,
Macquarie - MATH - 1802
Lecture XXIVMeasuresRemember that an elementary region R in E2 is a region that has a simple,closed, piecewise smooth curve C as its boundary. A regular region R is a regionthat is either regular or can be divided into nitely many regular regions. In
Macquarie - MATH - 1802
Lecture XXVGreens TheoremLet us dene a new type of derivative, called rotational derivative, applicable tovector elds in E2 . For such a vector eld F on a domain D in E2 , let us denethe rotational derivative at interior points of D. Here, a point P i
Macquarie - MATH - 1802
Lecture XXVIThe Divergence TheoremIn this lecture, we will dene a new type of derivative for vector elds on E3 ,called divergence. Let F be a vector eld dened on a domain D. Let us startby dening the divergence of F on interior points of D, i.e. point
Macquarie - MATH - 1802
Lecture XXVIIStokess TheoremIn the previous lecture, we saw how Greens theorem deals with integrals onelementary regions in E2 and their boundaries. We also saw how the divergencethoerem deals with elementary regions in E3 and their boundaries. We wil
Macquarie - MATH - 1802
Lecture XXVIIIMeasures; Irrotational elds1Circulation and ux measuresLet us rst see what the theorems from the past lectures say about the circulation and ux measures. From Greens theorem, we get that, if F is a C 1 vectoreld on D in E2 , thenF (R)
Macquarie - MATH - 1802
Lecture XXIXMathematical Applications1Leibnitzs RuleLeibnitzs Rule : Let f (x, t) be a C 1 function dened for a x b. Thenbdbff (x, t)dx =ft (x, t)dx, where ft (x, t) =.dt ataIn other words, if we dene g (t) =bAf (x, t)dx, thendgdt=ba
Macquarie - MATH - 1802
Lecture XXXnVectors and Matrices1nVectorsWe dene En to be the ndimensional Euclidean space, and Rn to be the setof points in En . Hence Rn is the set of all ordered ntuples of real numbers(x1 , . . . , xn ). Ordered ntuples allow repetitions, i.e. x
Macquarie - MATH - 1802
Lecture XXXILinear Equation SystemsAs we saw in the previous lecture, we can multiply m n matrices by columnnvectors. Consider the rows of an m n matrix A to be nvectors:1 Crc11r 2 2 C c2 r r =C = thenAA = . , C . .......m Cr
Macquarie - MATH - 1802
Lecture XXXIIRow Reduction; Determinants1Row ReductionRecall the 3 elementary operations we will use to solve the system of equationsAX = D:() multiplying an equation by a nonzero scalar;( ) adding to an equation some multiple of a dierent equation
Macquarie - MATH - 1802
Lecture XXXIIIDeterminants; Matrix Algebra1DeterminantsFor a square matrix A, the determinant of A has the following properties:1. Interchanging two rows of the matrix multiplies the value of the determinant by 1.2. If there exists two identical ro
Macquarie - MATH - 1802
Lecture XXXIVSubspacesIn the previous lecture we have seen that there are two methods for ndingthe inverse of a square matrix A. In the rst method, we use the fact that if Ais nonsingular, then the rowreduced form of [A : I ] is [I : A1 ]. In the seco
Macquarie - MATH - 1802
Macquarie - MATH - 1802
Macquarie - MATH - 1802
Macquarie - MATH - 1802
Macquarie - MATH - 1802
Macquarie - MATH - 1802
Clarion - BIOCHEMIST - 301
Name_TF_Credit_1. A2. D3. E4. C5. B6. E7. A8. A9. D10. E11. C12. C13. A14. C15. D16. D17. D18. B19. A20. A21. (5pts)A) List two structural differences between DNA and RNA.Deoxyribose vs ribose, typically ds vs ss, presence of tymidi
Harvard - BIO - 100
bio
Harvard - BIO - 100
Harvard - BIO - 100
BIO LCV LAB EXAM REVIEWhttp:/www.kumc.edu/instruction/medicine/anatomy/histoweb/index.htmLab 1 Body organization4 types of tissues:EpithelialcoverssurfacesoShape: Squamous, cuboidal, columnar Layers : simple (secrete/absorb), pseudostratified (cili
Harvard - BIO - 100
Chapter44:RegulationofInternalEnvironment: Animalsmaintainhomeostasisevenwhenputinconditionstoxicforindividualcells 3functionsofhomeostasis1. Thermoregulationmaintainbodytemperature2. Osmoregulation(kidney)regulatesolutebalanceandgain/lossofH2O3. E
Harvard - BIO - 100
Chapter 46 Reproduction Sexual reproduction-formation of offspring by fusion of haploidgametes-ovumfemale gamete-spermmale gamete Reproductive anatomy male external structurespenisscrotum and-high temperature inhibits sperm production Interna
Harvard - BIO - 100
==
Harvard - BIO - 100
Harvard - BIO - 100
9
Harvard - BIO - 100
Harvard - BIO - 100
888
Harvard - BIO - 100
86
Harvard - BIO - 100
9
Harvard - BIO - 100
-only ~ 1/100th of eggs released in a lifetime Usually only one egg matures and isreleased during each menstrual cycleAfter ovulation, remaining follicular tissuedevelops into corpus luteum-secretes estrogens and progesterone-maintenance of uterine
Harvard - BIO - 100
Assignment 7CHE-LCUTo be handed in on Thursday April 21th at the beginning of classCarbocation Rearrangements1. a) A student performed a reaction between 2-bromo-3,3-dimethylpentane and potassium-tertbutoxide at an elevated temperature. He was expecti
Harvard - BIO - 100
Assignment 8CHE-LCUTo be handed in on Monday April 2nd at the beginning of classRoad Maps1. a) Provide structures for A-GAOsO4GBBr2HBrH2OH2O2H2OCBH3:THF+HFH2O2, OH-I2HBrDEb) Which letters correspond to Markovnikov additions?c) Whi
Harvard - BIO - 100
Assignment 9CHE-LCUNo Due DateThese are questions from your textbook, Organic Chemistry by Solomons, 10th edition. The equivalentquestions in the 9th edition are also posted. You can find most questions in the middle of the respectivechapter with exe
Harvard - BIO - 100
Cookbook Table of ContentsCHE-LCUReactionsPage1. SN2-Inversion of configuration-Racemization2. SN13. E1 and E2- Zaitsev vs Hofmann- Anti coplanar requirement4. Synthesis of Alkynes- Terminal vs nonterminal stoichiometry5. Alkynes as nucleop
Harvard - BIO - 100
Is it substitution or elimination?Take some of the guesswork out of your decision. This table, in conjunction with thenotes that follow, will help you decide.Methyl &amp;1oSN2Strong NHigh [N]2o3oSN2allylic &amp;benzylic(SN2)SN2Strong NHigh [N]Str
Harvard - BIO - 100
Assignment 9CHE-LCUNo Due DateThese are questions from your textbook, Organic Chemistry by Solomons, 10th edition. The equivalentquestions in the 9th edition are also posted. You can find most questions in the middle of the respectivechapter with exe