Introduction to Calculus Volume II by J.H. Heinbockel The regular solids or regular polyhedra are solid geometric figures with the same identical regular polygon on each face. There are only five regular solids discovered by the ancient Greek mathematicians. These five solids are the following. the tetrahedron (4 faces) the cube or hexadron (6 faces) the octahedron (8 faces) the dodecahedron (12 faces) the icosahedron (20 faces) Each figure followsthe Euler formula Number of faces + Number of vertices = Number of edges + 2 F + V = E + 2 Introduction to Calculus Volume II by J.H. Heinbockel Emeritus Professor of Mathematics Old Dominion University (cid:13)cCopyright 2012 by John H. Heinbockel All rights reserved Paper or electronic copies for noncommercial use may be made freely without explicit permission of the author. All other rights are reserved. This Introduction to Calculus is intended to be a free ebook where portions of the text can be printed out. Commercial sale of this book or any part of it is strictly forbidden. ii Preface This is the second volume of an introductory calculus presentation intended for future scientists and engineers. Volume II is a continuation of volume I and contains chapters six through twelve. The chapter six presents an introduction to vectors, vector operations, dif- ferentiationand integrationof vectorswithmanyapplication. The chapterseveninvestigates curves and surfaces represented in a vector form and examines vector operations associated with these forms. Also investigated are methods for representing arclength, surface area and volume elementsfrom vector representations. The directionalderivativeis defined along with other vector operations and their properties as these additional vectors enable one to find maximum and minimum values associated with functions of more than one variable. The chapter 8 investigates scalar and vector fields and operations involving these quantities. The Gauss divergence theorem, the Stokes theorem and Green’s theorem in the plane along with applications associated with these theorems are investigated in some detail. The chap- ter 9 presents applications of vectors from selected areas of science and engineering. The chapter 10 presents an introduction to the matrix calculus and the difference calculus. The chapter 11 presents an introduction to probability and statistics. The chapters 10 and 11 are presented because in todays society technology development is tending toward a digital world and students should be exposed to some of the operational calculus that is going to be needed in order to understand some of this technology. The chapter 12 is added as an after thought to introduce those interested into some more advanced areas of mathematics. If you are a beginner in calculus, then be sure that you have had the appropriate back- ground material of algebra and trigonometry. If you don’t understand something then don’t be afraid to ask your instructor a question. Go to the library and check out some other calculus books to get a presentation of the subject from a different perspective. The internet is a place where one can find numerous help aids for calculus. Also on the internet one can find many illustrations of the applications of calculus. These additional study aids will show you that there are multipleapproaches to various calculussubjects and should help you with the development of your analytical and reasoning skills. J.H. Heinbockel January 2016 iii Table of Contents Introduction to Calculus Volume II Chapter 6 Introduction to Vectors ......................................1 Vectors and Scalars, Vector Addition and Subtraction, Unit Vectors, Scalar or Dot Product(innerproduct),DirectionCosinesAssociatedwithVectors,ComponentForm for Dot Product, The Cross Product or Outer Product, Geometric Interpretation, Vector Identities, Moment Produced by a Force, Moment About Arbitrary Line, Differentiation of Vectors, Differentiation Formulas, Kinematics of Linear Motion, Tangent Vector to Curve, Angular Velocity, Two-Dimensional Curves, Scalar and Vector Fields, Partial Derivatives, Total Derivative, Notation, Gradient, Divergence and Curl,TaylorSeries forVector Functions, Differentiation of Composite Functions, Integration of Vectors, Line Integrals of Scalar and Vector Functions, Work Done, Representation of Line Integrals Chapter 7 Vector Calculus I .............................................81 Curves,TangentstoSpaceCurve,NormalandBinormaltoSpaceCurve,Surfaces,The Sphere, The Ellipsoid, The Elliptic Paraboloid, The Elliptic Cone, The Hyperboloid of One Sheet, The Hyperboloid of Two Sheets, The Hyperbolic Paraboloid, Surfaces of Revolution, Ruled Surfaces, Surface Area, Arc Length, The Gradient, Divergence and Curl, Properties of the Gradient, Divergence and Curl, Directional Derivatives, ApplicationsfortheGradient,MaximumandMinimumValues,LagrangeMultipliers, Generalization of Lagrange Multipliers, Vector Fields and Field Lines, Surface and VolumeIntegrals, Normalto a Surface, Tangent Plane to Surface, Element of Surface Area, Surface Placed in a Scalar Field, Surace Placed in a Vector Field, Summary, VolumeIntegrals,Other VolumeElements, CylindricalCoordinates(r,θ,z),Spherical Coordinates (ρ,θ,φ) iv Table of Contents Chapter 8 Vector Calculus II ................................................173 Vector Fields, Divergence of Vector Field, The Gauss Divergence Theorem, Physical Interpretation of Divergence, Green’s Theorem in the Plane, Area Inside Simple Closed Curve, Change of Variable in Green’s Theorem, The Curl of a Vector Field, Physical Interpretation of Curl, Stokes Theorem, Related Integral Theorems, Region of Integration, Green’s First and Second Identities, Additional Operators, Relations Involvingthe Del Operator, Vector Operators and Curvilinear Coordinates, Orthogonal Curvilinear Coordinates, Transformation of Vectors, General Coordinate Transformation, Gradient in a General Orthogonal SystemofCoordinates,DivergenceinaGeneralOrthogonalSystemofCoordinates, CurlinaGeneralOrthogonalSystemofCoordinates,TheLaplacianinGeneralized Orthogonal Coordinates Chapter 9 Applications of Vectors ........................................241 ApproximationofVectorField,SphericalTrigonometry,VelocityandAcceleration in Polar Coordinates, Velocity and Acceleration in CylindricalCoordinates, Velocity and Acceleration in Spherical Coordinates, Introduction to Potential Theory,SolenoidalFields,Two-DimensionalConservativeVectorFields,FieldLines and Orthogonal Trajectories, Vector Fields Irrotational and Solenoidal, Laplace’s Equation, Three-dimensional Representations, Two-dimensional Representations, One-dimensional Representations, Three-dimensional Conservative Vector Fields, Theory of Proportions, Method of Tangents, Solid Angles, Potential Theory, Velocity Fields and Fluids, Heat Conduction, Two-body Problem, Kepler’s Laws, Vector DifferentialEquations,Maxwell’sEquations, Electrostatics, Magnetostatics Chapter 10 Matrix and Difference Calculus ............................307 The Matrix Calculus, Properties of Matrices, Dot or Inner Product, Matrix Multiplica- tion, Special Square Matrices, The Inverse Matrix, Matrices with Special Properties, The Determinant of a Square Matrix, Minors and Cofactors, Properties of Determinants, Rank of a Matrix, Calculation of Inverse Matrix, Elementary Row Operations, Eigenvalues and Eigenvectors, Properties of Eigenvalues and Eigenvectors, Additional Properties Involv- ing Eigenvalues and Eigenvectors, Infinite Series of Square Matrices, The Hamilton-Cayley Theorem, Evaluationof Functions, Four-terminalNetworks, Calculus ofFinite Differences, Differences and Difference Equations, Special Differences, Finite Integrals, Summation of Series, Difference EquationswithConstant Coefficients,NonhomogeneousDifference Equa- tions v Table of Contents Chapter 11 Introduction to Probability and Statistics ...381 Introduction, Simulations,Representation of Data,Tabular Representation of Data, Arithmetic Mean or Sample Mean, Median, Mode and Percentiles, The Geometric and Harmonic Mean, The Root Mean Square (RMS), Mean Deviation and Sample Variance, Probability, Probability Fundamentals, Probability of an Event, Conditional Probability, Permutations, Combinations, Binomial Coefficients, Discrete and Continuous Probability Distributions, Scaling, The Normal Distribution, Standardization, The Binomial Distribu- tion,The MultinomialDistribution,The Poisson Distribution,The Hypergeometric Distri- bution, The Exponential Distribution,The GammaDistribution, Chi-Square Distribution, Student’s t-Distribution, The F-Distribution, The Uniform Distribution, Confidence Inter- vals, Least Squares Curve Fitting, Linear Regression, Monte Carlo Methods, Obtaining a Uniform Random Number Generator, Linear Interpolation Chapter 12 Introduction to more Advanced Material.....449 An integrationmethod, The use of integrationto sum infiniteseries, Refractionthrough a prism,DifferentiationofImplicitFunctions,oneequationtwounknowns,oneequationthree unknowns, one equation four unknowns, one equation n-unknowns, two equations three unknowns, two equations four unknowns, three equations five unknowns, Generalization, The Gamma function, Product of odd and even integers, Various representations for the Gammafunction,Euler formulafor the Gammafunction,The Zeta function related to the Gammafunction,Product property ofthe Gammafunction,Derivatives of lnΓ(z) ,Taylor seriesexpansionforlnΓ(x+1),Anotherproductformula,Useofdifferentialequationstofind −1 series, The Laplace Transform, Inverse Laplace transform L , Properties of the Laplace transform, Introduction to Complex Variable Theory, Derivative of a Complex Function, Integration of a Complex Function, Contour integration, Indefinite integration, Definite integrals, Closed curves, The Laurent series Appendix A Units of Measurement ................................510 Appendix B Background Material ..................................512 Appendix C Table of Integrals .......................................524 Appendix D Solutions to Selected Problems ...................578 Index ............................................................................619 vi 1 Chapter 6 Introduction to Vectors Scalars are quantities with magnitude only whereas vectors are those quantities having both a magnitude and a direction. Vectors are used to model a variety of fundamental processes occurring in engineering, physics and the sciences. The material presented in the pages that follow investigates both scalar and vectors quantities and operations associated with their use in solving applied problems. In particular, differentiationandintegrationtechniquesassociated with both scalar and vector quantities will be investigated. Vectors and Scalars A vector is any quantity which possesses both magnitude and direction. A scalar is a quantity which possesses a magnitude but does not possess a direction. Examples of vector quantities are force, velocity, acceleration, momentum, weight, torque, angular velocity, angular acceleration, angular momentum. Examples of scalar quantities are time, temperature, size of an angle, energy, mass, length, speed, density A vector can be represented by an arrow. The orientation of the arrow determines the direction of the vector, and the length of the arrow is associated with the magnitude of the vector. The magnitude of a vector B(cid:1) is denoted | B(cid:1) | or B and represents the length of the vector. The tail end of the arrow is called the origin, and the arrowhead is called the terminus. Vectors are usually denoted by letters in Figure 6-1. bold face type. When a bold face type is inconve- Scalar multiplication. nient to use, then a letter with an arrow over it is employed, such as, A(cid:1), B(cid:1), C(cid:1). Throughout this text the arrow notation is used in all discussions of vectors. Properties of Vectors Some important properties of vectors are 1. Two vectors A(cid:1) and B(cid:1) are equal if they have the same magnitude (length) and direction. Equality is denoted by A(cid:1) =B(cid:1). 2 2. The magnitude of a vector is a nonnegative scalar quantity. The magnitude of a vector B(cid:1) is denoted by the symbols B or |B(cid:1)|. 3. A vector B(cid:1) is equal to zero only if its magnitude is zero. A vector whose mag- nitude is zero is called the zero or null vector and denoted by the symbol (cid:1)0. 4. Multiplication of a nonzero vector B(cid:1) by a positive scalar m is denoted by mB(cid:1) and produces a new vector whose direction is the same as B(cid:1) but whose magnitude is m times the magnitude of B(cid:1). Symbolically, |mB(cid:1)|=m|B(cid:1)|. If m is a negative scalar the direction of mB(cid:1) is opposite to that of the direction of B(cid:1). In figure 6-1 several vectors obtained from B(cid:1) by scalar multiplication are exhibited. 5. Vectors are considered as “free vectors”. The term “free vector” is used to mean the following. Any vector may be moved to a new position in space provided that in the new position it is parallel to and has the same direction as its original position. In many of the examples that follow, there are times when a given vector is moved to a convenient point in space in order to emphasize a special geometrical or physical concept. See for example figure 6-1. Vector Addition and Subtraction Let C(cid:1) = A(cid:1) +B(cid:1) denote the sum of two vectors A(cid:1) and B(cid:1). To find the vector sum A(cid:1)+B(cid:1), slidethe originof the vectorB(cid:1) to the terminuspoint of the vectorA(cid:1), thendraw the line from the origin of A(cid:1) to the terminus of B(cid:1) to representC(cid:1). Alternatively, start with the vector B(cid:1) and place the origin of the vector A(cid:1) at the terminus point of B(cid:1) to constructthe vector B(cid:1)+A(cid:1). Adding vectors in this way employsthe parallelogramlaw for vector addition which is illustrated in the figure 6-2. Note that vector addition is commutative. That is, using the shifted vectors A(cid:1) and B(cid:1), as illustrated in the figure 6-2, the commutative law for vector addition A(cid:1) +B(cid:1) =B(cid:1) +A(cid:1), is illustrated using the parallelogram illustrated. The addition of vectors can be thought of as connecting the origin and terminus of directed line segments. Figure 6-2. Parallelogram law for vector addition