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Introduction to Calculus Volume II by J.H. Heinbockel



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Introduction to Calculus Volume II written by J.H. Heinbockel , Emeritus Professor of Mathematics, Old Dominion University. 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, differentiation and integration of vectors with many application. The chapter seven investigates 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 elements from vector representations. The directional derivative is 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 chapter 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.

Introduction to Calculus Volume II written by J.H. Heinbockel cover the following topics.

  • 6. Introduction to Vectors Vectors and Scalars, Vector Addition and Subtraction, Unit Vectors, Scalar or Dot Product (inner product), Direction Cosines Associated with Vectors, Component Form 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, Taylor Series for Vector Functions, Differentiation of Composite Functions, Integration of Vectors, Line Integrals of Scalar and Vector Functions, Work Done, Representation of Line Integrals

  • 7. Vector Calculus I Curves, Tangents to Space Curve, Normal and Binormal to Space Curve, 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, Applications for the Gradient, Maximum and Minimum Values, Lagrange Multipliers, Generalization of Lagrange Multipliers, Vector Fields and Field Lines, Surface and Volume Integrals, Normal to a Surface, Tangent Plane to Surface, Element of Surface Area, Surface Placed in a Scalar Field, Surace Placed in a Vector Field, Summary, Volume Integrals, Other Volume Elements, Cylindrical Coordinates (r, θ, z), Spherical Coordinates (ρ, θ, φ)

  • 8. Vector Calculus II 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 Involving the Del Operator, Vector Operators and Curvilinear Coordinates, Orthogonal Curvilinear Coordinates, Transformation of Vectors, General Coordinate Transformation, Gradient in a General Orthogonal System of Coordinates, Divergence in a General Orthogonal System of Coordinates, Curl in a General Orthogonal System of Coordinates, The Laplacian in Generalized Orthogonal Coordinates

  • 9. Applications of Vectors Approximation of Vector Field, Spherical Trigonometry, Velocity and Acceleration in Polar Coordinates, Velocity and Acceleration in Cylindrical Coordinates, Velocity and Acceleration in Spherical Coordinates, Introduction to Potential Theory, Solenoidal Fields, Two-Dimensional Conservative Vector Fields, Field Lines 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 Differential Equations, Maxwell’s Equations, Electrostatics, Magnetostatics

  • 10. Matrix and Difference Calculus The Matrix Calculus, Properties of Matrices, Dot or Inner Product, Matrix Multiplication, 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 Involving Eigenvalues and Eigenvectors, Infinite Series of Square Matrices, The Hamilton-Cayley Theorem, Evaluation of Functions, Four-terminal Networks, Calculus of Finite Differences, Differences and Difference Equations, Special Differences, Finite Integrals, Summation of Series, Difference Equations with Constant Coefficients, Nonhomogeneous Difference Equations

  • 11. Introduction to Probability and Statistics 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 Distribution, The Multinomial Distribution, The Poisson Distribution, The Hypergeometric Distribution, The Exponential Distribution, The Gamma Distribution, Chi-Square Distribution, Student’s t-Distribution, The F-Distribution, The Uniform Distribution, Confidence Intervals, Least Squares Curve Fitting, Linear Regression, Monte Carlo Methods, Obtaining a Uniform Random Number Generator, Linear Interpolation

  • 12. Introduction to more Advanced Material An integration method, The use of integration to sum infinite series, Refraction through a prism, Differentiation of Implicit Functions, one equation two unknowns, one equation three 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 Gamma function, Euler formula for the Gamma function, The Zeta function related to the Gamma function, Product property of the Gamma function, Derivatives of ln Γ(z) , Taylor series expansion for ln Γ(x+1), Another product formula, Use of differential equations to find series, The Laplace Transform, Inverse Laplace transform L−1, 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
    B Background Material
    C Table of Integrals
    D Solutions to Selected Problems

  • Index

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