We now begin with vector calculus which concerns two kinds of functions: Vector functions: Functions whose values are vectors depending on the points
|
|
- Baldwin Pope
- 5 years ago
- Views:
Transcription
1 94 Vector and scalar functions and fields Derivatives We now begin with vector calculus which concerns two kinds of functions: Vector functions: Functions whose values are vectors depending on the points v v( ) v ( ) iv ( ) jv ( ) k in space, Scalar functions: Functions whose values are scalars depending on the points in space, f f( ) Here, is a point in the domain of definition, which in applications is a 3-D domain or a surface or a curve in space A vector function defines a vector field and a scalar function defines a scalar field in that domain or on that surface or curve Examples of vector fields are field of tangent vectors of a curve, field of normal vectors of a surface, velocity field of a rotating body and the gravitational field (see Figs ) Examples of scalar fields are the temperature field in a body or the pressure field of the air in the earth's atmosphere Vector and scalar fields may also depend on time t or on some other parameters If we introduce Cartesian coordinates x, yz,, then instead of v( ) we can write, v v( x, yz, ) v( xyz,, ) iv( xyz,, ) jv( xyz,, ) k Note that the components depend on the choice of the coordinate system, whereas a vector field that has a physical or a geometric meaning should have magnitude and direction depending only on, not on the choice of coordinate system Similarly for the value of a scalar field f ( ) f( x, y, z) Example 1: Scalar function (Euclidean distance in space) The distance f ( ) of any point from a fixed point in space is a scalar function whose domain of definition is the whole space f ( ) defines a scalar field in space If a Cartesian coordinate system is introduced and has the coordinates x, y, z and has the coordinates x, yz, then f is given by, f( ) f( x, y, z) [( xx ) ( y y ) ( zz ) ] 1/ If the given Cartesian coordinate system is replaced by another Cartesian coordinate system obtained by translating and rotating the given system, then the values of the coordinates of and will in general change, but f ( ) will have the same value as before Hence f ( ) is a scalar function June 5,
2 Example : Vector field (Velocity field) At any instant the velocity vectors v( ) of a rotating body B constitutes a vector field, called the velocity field of the rotation If a Cartesian coordinate system having the origin on the axis of rotation is introduced then (see Ex 5, Sec 93), v( x, yz, ) wr w( xi yj zk ), (941) where x, yz, are the coordinates of any point of B at the instant under consideration If the coordinate system is such that the z -axis is the axis of rotation and w points in the positive z -direction, then w k and i j k v ( yixj ) x y z An example of a rotating body and the corresponding velocity field are shown in Fig 195 Example 3: Vector field (Field of force, gravitational field) A particle A of mass M is fixed to a point and a particle B of mass m is free to take up various positions in space Then particle A attracts particle B According to Newton's law of gravitation the corresponding gravitational force p is directed from to, and its magnitude is proportional to 1/r, where r is the distance between and, c p, (94) r 8 3 where c GMm and G 6671 cm /(gm sec ) is the gravitational constant Hence p defines a vector field in space If a Cartesian coordinate system is introduced such that has the coordinates x, y, z and has the coordinates x, yz, then, r [( xx ) ( y y ) ( zz ) ] 1/ Assuming that r and introducing the vector, r ( x x ) i( y y ) j( zz ) k, June 5,
3 it is seen that r r and ( 1/ r) r is a unit vector in the direction of p ; the minus sign indicated that p is directed from to From this and eqn (94), 1 c c p p ( ) r r [( xx 3 3 ) i( yy) j( zz ) k ] (943) r r r This vector function describes the gravitational force acting on B 941 Vector calculus Next it is shown that the basic concepts of calculus, ie, convergence, continuity, and differentiability can be defined for vector functions in a simple and natural way Most important here is the derivative Convergence: An infinite sequence of vectors a ( n ), n 1,,, is said to converge if there is a vector a such that lim a a (944) n ( n) Here a is called the limit vector of that sequence, and we write, lima a (945) n ( n ) Cartesian coordinates being given, this sequence of vectors converge to a if and only if the three sequences of components of the vectors converge to the corresponding components of a Similarly, a vector function v( t) of a real variable t is said to have the limit l as t approaches t, if ( t) is defined in some neighborhood of (possibly except at t ) and v t lim v( t) l (946) tt Then we write, lim v( t) l (947) tt Here a neighborhood of t is an interval (segment) on the t -axis containing t as an interior point (not as an endpoint) June 5,
4 Continuity: A vector function v() t is said to be continuous at t t if it is defined in some neighborhood of (including at t itself) and t lim v( t) v ( t ) (948) tt If a Cartesian coordinate system is introduced, we may write, v() t v () t iv () t jv () t k Then () t is continuous at if and only if its three components are continuous at t v t Derivative of a vector function: A vector function the following limit exists, v() t is said to be differentiable at a point t if v() t lim v( tt) v( t t This vector v ( t) is called the derivative of v( t) See fig 197 t ) (949) In components with respect to a given Cartesian coordinate system, v() t v() t iv() t jv() t k (941) Hence the derivative v ( t) is obtained by differentiating each component separately For instance, if v() t tit j then v( t) itj Equation (941) follows from (949) and conversely because (949) is a "vector form" of the usual formula for calculus by which the derivative of a function of a single variable is defined (The curve in Fig 197 is the locus of the terminal points representing v( t) for values of the independent variable in some interval containing t and t t in (949)) The familiar differentiation rules continue to hold for differentaiating vector functions, for instance, and in particular, ( cv) cv, uv, u vuv, (9411) uvuv, (941) u( vw) u( vw) u( vw) u( vw ) (9413) June 5,
5 Example 4: Derivative of a vector function of constant length Let v( t) be a vector function whose length is constant, say, v() t c Then v v v c, and ( vv) vv, by differentiation Hence, the derivative of a vector function v() t of constant length is either the zero vector or is perpendicular to v() t 94 artial derivatives of a vector function artial differentiation of vector functions of two or more variables can be introduced as follows Suppose that the components of a vector function, v() t v () t iv () t jv () t k, are differentiable functions of n variables t,, 1 tn Then the partial derivative of v with respect to t is denoted by v m / t m and is defined as the vector function, v v1 v v3 i j k t t t t m m m m Similarly, second partial derivatives are and so on v v1 v v3 i j k, t t t t t t t t l m l m l m l m Example 5: artial derivatives r Let r( t1, t) acost1iasint1j tk Then asin t1i acost1j and t 1 r t k Various physical and geometric applications of derivatives of vector functions will be discussed in the next sections and in Chapter 1 June 5,
CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS
CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS 4.1 Vector Functions 4.2 Calculus of Vector Functions 4.3 Tangents REVIEW: Vectors Scalar a quantity only with its magnitude Example: temperature, speed, mass, volume
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationMA102: Multivariable Calculus
MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationIntroduction to Vector Functions
Introduction to Vector Functions Differentiation and Integration Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction In this section, we study the differentiation
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationMaxima and Minima. (a, b) of R if
Maxima and Minima Definition Let R be any region on the xy-plane, a function f (x, y) attains its absolute or global, maximum value M on R at the point (a, b) of R if (i) f (x, y) M for all points (x,
More informationArnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VectorCalculus1 due 05/03/2008 at 02:00am EDT.
Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment Vectoralculus due 05/03/008 at 0:00am EDT.. ( pt) rochesterlibrary/setvectoralculus/ur V.pg onsider the transformation T : x = 35 35 37u
More informationThe Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE
36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"
More informationLOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort
- 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationOverview of vector calculus. Coordinate systems in space. Distance formula. (Sec. 12.1)
Math 20C Multivariable Calculus Lecture 1 1 Coordinates in space Slide 1 Overview of vector calculus. Coordinate systems in space. Distance formula. (Sec. 12.1) Vector calculus studies derivatives and
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More informationVECTORS IN A STRAIGHT LINE
A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II.
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationCourse 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationLecture notes: Introduction to Partial Differential Equations
Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential
More informationMechanics 1: Coordinates
Mechanics 1: Coordinates We have emphasized that our development of the definition and properties of vectors did not utilize at all the notion of a coordinate system, i.e., an agreed upon set of locations
More informationCh 4 Differentiation
Ch 1 Partial fractions Ch 6 Integration Ch 2 Coordinate geometry C4 Ch 5 Vectors Ch 3 The binomial expansion Ch 4 Differentiation Chapter 1 Partial fractions We can add (or take away) two fractions only
More informationThe Calculus of Vec- tors
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 3 1 The Calculus of Vec- Summary: tors 1. Calculus of Vectors: Limits and Derivatives 2. Parametric representation of Curves r(t) = [x(t), y(t),
More informationIntroduction to Vector Functions
Introduction to Vector Functions Limits and Continuity Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Introduction to Vector Functions Spring 2012 1 / 14 Introduction In this section, we study
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 EE2007:
More informationvector calculus 1 Learning outcomes
8 Contents vector calculus. Background to Vector Calculus. Differential Vector Calculus 3. Orthogonal curvilinear coordinates Learning outcomes In this Workbook, you will learn about scalar and vector
More informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More informationCulminating Review for Vectors
Culminating Review for Vectors 0011 0010 1010 1101 0001 0100 1011 An Introduction to Vectors Applications of Vectors Equations of Lines and Planes 4 12 Relationships between Points, Lines and Planes An
More informationDifferential Vector Calculus
Contents 8 Differential Vector Calculus 8. Background to Vector Calculus 8. Differential Vector Calculus 7 8.3 Orthogonal Curvilinear Coordinates 37 Learning outcomes In this Workbook you will learn about
More information4.1 Distance and Length
Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors
More informationM E 320 Professor John M. Cimbala Lecture 10
M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 My part:
More information16.3 Conservative Vector Fields
16.3 Conservative Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.3 Conservative Vector Fields M273, Fall 2011 1 / 23 Fundamental Theorem for Conservative Vector
More informationMath 265H: Calculus III Practice Midterm II: Fall 2014
Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question
More information(read nabla or del) is defined by, k. (9.7.1*)
9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationAppendix E : Note on regular curves in Euclidean spaces
Appendix E : Note on regular curves in Euclidean spaces In Section III.5 of the course notes we posed the following question: Suppose that U is a connected open subset of R n and x, y U. Is there a continuous
More informationLinear Algebra. 1.1 Introduction to vectors 1.2 Lengths and dot products. January 28th, 2013 Math 301. Monday, January 28, 13
Linear Algebra 1.1 Introduction to vectors 1.2 Lengths and dot products January 28th, 2013 Math 301 Notation for linear systems 12w +4x + 23y +9z =0 2u + v +5w 2x +2y +8z =1 5u + v 6w +2x +4y z =6 8u 4v
More informationMathematics Revision Questions for the University of Bristol School of Physics
Mathematics Revision Questions for the University of Bristol School of Physics You will not be surprised to find you have to use a lot of maths in your stu of physics at university! You need to be completely
More information12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes
Differentiation of vectors 12.5 Introuction The area known as vector calculus is use to moel mathematically a vast range of engineering phenomena incluing electrostatics, electromagnetic fiels, air flow
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationTopic Learning Outcomes Suggested Teaching Activities Resources On-Line Resources
UNIT 3 Trigonometry and Vectors (P1) Recommended Prior Knowledge. Students will need an understanding and proficiency in the algebraic techniques from either O Level Mathematics or IGCSE Mathematics. Context.
More informationLecture 11: Differential Geometry
Lecture 11: Differential Geometry c Bryan S. Morse, Brigham Young University, 1998 2000 Last modified on February 28, 2000 at 8:45 PM Contents 11.1 Introduction..............................................
More informationOWELL WEEKLY JOURNAL
Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --
More informationA Note on the Barycentric Square Roots of Kiepert Perspectors
Forum Geometricorum Volume 6 (2006) 263 268. FORUM GEOM ISSN 1534-1178 Note on the arycentric Square Roots of Kiepert erspectors Khoa Lu Nguyen bstract. Let be an interior point of a given triangle. We
More informationSec. 1.1: Basics of Vectors
Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x
More information1 The Lagrange Equations of Motion
1 The Lagrange Equations of Motion 1.1 Introduction A knowledge of the rudiments of dynamics is essential to understanding structural dynamics. Thus this chapter reviews the basic theorems of dynamics
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2a-22 Ver: August 28, 2010 Ver 1.6: Martin Adams, Sep 2009 Ver 1.5: Martin Adams, August 2008 Ver
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationMOTION IN TWO OR THREE DIMENSIONS
MOTION IN TWO OR THREE DIMENSIONS 3 Sections Covered 3.1 : Position & velocity vectors 3.2 : The acceleration vector 3.3 : Projectile motion 3.4 : Motion in a circle 3.5 : Relative velocity 3.1 Position
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationMATH 19520/51 Class 5
MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential
More informationChapter 4: Fluid Kinematics
Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More informationCHAPTER 7 DIV, GRAD, AND CURL
CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationSection 4.3 Vector Fields
Section 4.3 Vector Fields DEFINITION: A vector field in R n is a map F : A R n R n that assigns to each point x in its domain A a vector F(x). If n = 2, F is called a vector field in the plane, and if
More informationARNOLD PIZER rochester problib from CVS Summer 2003
ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.
More information3. Interpret the graph of x = 1 in the contexts of (a) a number line (b) 2-space (c) 3-space
MA2: Prepared by Dr. Archara Pacheenburawana Exercise Chapter 3 Exercise 3.. A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and
More information3-D Kinetics of Rigid Bodies
3-D Kinetics of Rigid Bodies Angular Momentum Generalized Newton s second law for the motion for a 3-D mass system Moment eqn for 3-D motion will be different than that obtained for plane motion Consider
More information9.4 Vector and Scalar Fields; Derivatives
9.4 Vector and Scalar Fields; Derivatives Vector fields A vector field v is a vector-valued function defined on some domain of R 2 or R 3. Thus if D is a subset of R 3, a vector field v with domain D associates
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.
More informationWe know how to identify the location of a point by means of coordinates: (x, y) for a point in R 2, or (x, y,z) for a point in R 3.
Vectors We know how to identify the location of a point by means of coordinates: (x, y) for a point in R 2, or (x, y,z) for a point in R 3. More generally, n-dimensional real Euclidean space R n is the
More informationStudent name: MATH 350: Multivariate Calculus March 30, 2018 Test 2 - In class part
Student name: MATH 350: Multivariate Calculus March 30, 2018 Test 2 - In class part Instructions: Answer all questions on separate paper (not on this sheet!). The only reference materials allowed on this
More informationW i n t e r r e m e m b e r t h e W O O L L E N S. W rite to the M anageress RIDGE LAUNDRY, ST. H E LE N S. A uction Sale.
> 7? 8 «> ««0? [ -! ««! > - ««>« ------------ - 7 7 7 = - Q9 8 7 ) [ } Q ««
More informationr/lt.i Ml s." ifcr ' W ATI II. The fnncrnl.icniccs of Mr*. John We mil uppn our tcpiiblicnn rcprc Died.
$ / / - (\ \ - ) # -/ ( - ( [ & - - - - \ - - ( - - - - & - ( ( / - ( \) Q & - - { Q ( - & - ( & q \ ( - ) Q - - # & - - - & - - - $ - 6 - & # - - - & -- - - - & 9 & q - / \ / - - - -)- - ( - - 9 - - -
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationII. An Application of Derivatives: Optimization
Anne Sibert Autumn 2013 II. An Application of Derivatives: Optimization In this section we consider an important application of derivatives: finding the minimum and maximum points. This has important applications
More information6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant
DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional
More informationMATH1012 Mathematics for Civil and Environmental Engineering Prof. Janne Ruostekoski School of Mathematics University of Southampton
MATH02 Mathematics for Civil and Environmental Engineering 20 Prof. Janne Ruostekoski School of Mathematics University of Southampton September 25, 20 2 CONTENTS Contents Matrices. Why matrices?..................................2
More informationLecture Notes for MATH2230. Neil Ramsamooj
Lecture Notes for MATH3 Neil amsamooj Table of contents Vector Calculus................................................ 5. Parametric curves and arc length...................................... 5. eview
More informationVector valued functions
98 Chapter 3 Vector valued functions In this chapter we study two types of special functions: (1) Continuous mapping of one variable(called a path) (2) Mapping from a subset of R n to itself(called vector
More informationMATH107 Vectors and Matrices
School of Mathematics, KSU 20/11/16 Vector valued functions Let D be a set of real numbers D R. A vector-valued functions r with domain D is a correspondence that assigns to each number t in D exactly
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More informationVector Geometry. Chapter 5
Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at
More informationSome common examples of vector fields: wind shear off an object, gravitational fields, electric and magnetic fields, etc
Vector Analysis Vector Fields Suppose a region in the plane or space is occupied by a moving fluid such as air or water. Imagine this fluid is made up of a very large number of particles that at any instant
More informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete
More informationVector Algebra August 2013
Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented
More informationChapter 7. Kinetic energy and work. Energy is a scalar quantity associated with the state (or condition) of one or more objects.
Chapter 7 Kinetic energy and work 7.2 What is energy? One definition: Energy is a scalar quantity associated with the state (or condition) of one or more objects. Some characteristics: 1.Energy can be
More informationVector Calculus - GATE Study Material in PDF
Vector Calculus - GATE Study Material in PDF In previous articles, we have already seen the basics of Calculus Differentiation and Integration and applications. In GATE 2018 Study Notes, we will be introduced
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
More informationRoot test. Root test Consider the limit L = lim n a n, suppose it exists. L < 1. L > 1 (including L = ) L = 1 the test is inconclusive.
Root test Root test n Consider the limit L = lim n a n, suppose it exists. L < 1 a n is absolutely convergent (thus convergent); L > 1 (including L = ) a n is divergent L = 1 the test is inconclusive.
More informationIntroduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationAPPLICATIONS OF INTEGRATION
6 APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6.5 Average Value of a Function In this section, we will learn about: Applying integration to find out the average value of a function. AVERAGE
More informationINTEGRAL CALCULUS DIFFERENTIATION UNDER THE INTEGRAL SIGN: Consider an integral involving one parameter and denote it as
INTEGRAL CALCULUS DIFFERENTIATION UNDER THE INTEGRAL SIGN: Consider an integral involving one parameter and denote it as, where a and b may be constants or functions of. To find the derivative of when
More informationa b = a a a and that has been used here. ( )
Review Eercise ( i j+ k) ( i+ j k) i j k = = i j+ k (( ) ( ) ) (( ) ( ) ) (( ) ( ) ) = i j+ k = ( ) i ( ( )) j+ ( ) k = j k Hence ( ) ( i j+ k) ( i+ j k) = ( ) + ( ) = 8 = Formulae for finding the vector
More informationChapter 14: Vector Calculus
Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationElements of Vector Calculus : Line and Surface Integrals
Elements of Vector Calculus : Line and Surface Integrals Lecture 2: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay In this lecture we will talk about special functions
More informationLOWELL WEEKLY JOURNAL
: Y J G V $ 5 V V G Y 2 25 Y 2» 5 X # VG q q q 6 6 X J 6 $3 ( 6 2 6 2 6 25 3 2 6 Y q 2 25: JJ JJ < X Q V J J Y J Q V (» Y V X Y? G # V Y J J J G J»Y ) J J / J Y Y X ({ G #? J Y ~» 9? ) < ( J VY Y J G (
More informationMIDTERM EXAMINATION. Spring MTH301- Calculus II (Session - 3)
ASSALAM O ALAIKUM All Dear fellows ALL IN ONE MTH3 Calculus II Midterm solved papers Created BY Ali Shah From Sahiwal BSCS th semester alaoudin.bukhari@gmail.com Remember me in your prayers MIDTERM EXAMINATION
More information