Essential Engineering Mathematics

Michael Batty and Ventus Publishing ApS, 2011. — 149 p. This short text aims to be somewhere first to look to refresh your mathematical techniques and remind uoy, of some of the principles behind them. This book includes such topics as vectors and matrices, functions and limits, calculus of one and many variables, ordinary differential equations, and complex function theory. Content 0. Introduction 1. Preliminaries 1.1 Number Systems: The Integers, Rationals and Reals 1.2 Working with the Real Numbers 1.2.1 Intervals 1.2.2 Solving Inequalities 1.2.3 Absolute Value 1.2.4 Inequalities Involving Absolute Value 1.3 Complex Numbers 1.3.1 Imaginary Numbers 1.3.2 The Complex Number System and its Arithmetic 1.3.3 Solving Polynomial Equations Using Complex Numbers 1.3.4 Geometry of Complex Numbers 2. Vectors and Matrices 2.1 Vectors 2.2 Matrices and Determinants 2.2.1 Arithmetic of Matrices 2.2.2 Inverse Matrices and Determinants 2.2.3 The Cross Product 2.3 Systems of Linear Equations and Row Reduction 2.3.1 Systems of Linear Equations 2.3.2 Row Reduction 2.3.3 Finding the Inverse of a Matrix using Row Reduction 2.4 Bases 2.5 Eigenvalues and Eigenvectors 3. Functions and Limits 3.1 Functions 3.1.1 Denition of a Function 3.1.2 Piping Functions Together 3.1.3 Inverse Functions 3.2 Limits 3.3 Continuity 4. Calculus of One Variable Part 1: Differentiation 4.1 Derivatives 4.2 The Chain Rule 4.3 Some Standard Derivatives 4.4 Dierentiating Inverse Functions 4.5 Implicit Differentiation 4.6 Logarithmic Differentiation 4.7 Higher Derivatives 4.8 L’Hôpital’s Rule 4.9 Taylor Series 5. Calculus of One Variable Part 2: Integration 5.1 Summing Series 5.2 Integrals 5.3 Antiderivatives 5.4 Integration by Substitution 5.5 Partial Fractions 5.6 Integration by Parts 5.7 Reduction Formulae 5.8 Improper Integrals 6. Calculus of Many Variables 6.1 Surfaces and Partial Derivatives 6.2 Scalar Fields 6.3 Vector Fields 6.4 Jacobians and the Chain Rule 6.5 Line Integrals 6.6 Surface and Volume Integrals 7. Ordinary Differential Equations 7.1 First Order Dierential Equations Solvable by Integrating Factor 7.2 First Order Separable Differential Equations 7.3 Second Order Linear Differential Equations with Constant Coefficients: The Homogeneous Case 7.4 Second Order Linear Differential Equations with Constant Coefficients: The Inhomogeneous Case 7.5 Initial Value Problems 8. Complex Function Theory 8.1 Standard Complex Functions 8.2 The Cauchy-Riemann Equations 8.3 Complex Integrals Index Sisältö Preliminaries Number Systems: The Integers, Rationals and Reals Working with the Real Numbers Complex Numbers Vectors and Matrices Vectors Matrices and Determinants Systems of Linear Equations and Row Reduction Bases Eigenvalues and Eigenvectors Functions and Limits Functions Limits Continuity Calculus of One Variable Part 1: Differentiation Derivatives The Chain Rule Some Standard Derivatives Dierentiating Inverse Functions Implicit Differentiation Logarithmic Differentiation Higher Derivatives L’Hôpital’s Rule Taylor Series Calculus of One Variable Part 2: Integration Summing Series Integrals Antiderivatives Integration by Substitution Partial Fractions Integration by Parts Reduction Formulae Improper Integrals Calculus of Many Variables Surfaces and Partial Derivatives Scalar Fields Vector Fields Jacobians and the Chain Rule Line Integrals Surface and Volume Integrals Ordinary Differential Equations First Order Dierential Equations Solvable by Integrating Factor First Order Separable Differential Equations Second Order Linear Differential Equations with Constant Coefficients: The Homogeneous Case Second Order Linear Differential Equations with Constant Coefficients: The Inhomogeneous Case Initial Value Problems Complex Function Theory Standard Complex Functions The Cauchy-Riemann Equations Complex Integrals Index This textbook covers topics such as functions, single variable calculus, multivariate calculus, differential equations and complex functions. The necessary linear algebra for multivariate calculus is also outlined. More advanced topics which have been omitted, but which you will certainly come across, are partial differential equations, Fourier transforms and Laplace transforms. You can download the book for free via the link below.