Advanced Engineering Mathematics

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Cover; Title Page; Copyright Page; Contents; Preface; Part One: Review Material; Chapter 1. Review of Prerequisites; 1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions; 1.2 Complex Numbers; 1.3 The Complex Plane; 1.4 Modulus and Argument Representation of Complex Numbers; 1.5 Roots of Complex Numbers; 1.6 Partial Fractions; 1.7 Fundamentals of Determinants; 1.8 Continuity in One or More Variables; 1.9 Differentiability of Functions of One or More Variables; 1.10 Tangent Line and Tangent Plane Approximations to Functions; 1.11 Integrals; 1.12 Taylor and Maclaurin Theorems. 1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation1.14 Inverse Functions and the Inverse Function Theorem; Part Two: Vectors and Matrices; Chapter 2. Vectors and Vector Spaces; 2.1 Vectors, Geometry, and Algebra; 2.2 The Dot Product (Scalar Product); 2.3 The Cross Product (Vector Product); 2.4 Linear Dependence and Independence of Vectors and Triple Products; 2.5 n-Vectors and the Vector Space Rn; 2.6 Linear Independence, Basis, and Dimension; 2.7 Gram-Schmidt Orthogonalization Process; Chapter 3. Matrices and Systems of Linear Equations. 3.1 Matrices3.2 Some Problems That Give Rise to Matrices; 3.3 Determinants; 3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication; 3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix; 3.6 Row and Column Spaces and Rank; 3.7 The Solution of Homogeneous Systems of Linear Equations; 3.8 The Solution of Nonhomogeneous Systems of Linear Equations; 3.9 The Inverse Matrix; 3.10 Derivative of a Matrix; Chapter 4. Eigenvalues, Eigenvectors, and Diagonalization; 4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors. 4.2 Diagonalization of Matrices4.3 Special Matrices with Complex Elements; 4.4 Quadratic Forms; 4.5 The Matrix Exponential; Part Three: Ordinary Differential Equations; Chapter 5. First Order Differential Equations; 5.1 Background to Ordinary Differential Equations; 5.2 Some Problems Leading to Ordinary Differential Equations; 5.3 Direction Fields; 5.4 Separable Equations; 5.5 Homogeneous Equations; 5.6 Exact Equations; 5.7 Linear First Order Equations; 5.8 The Bernoulli Equation; 5.9 The Riccati Equation; 5.10 Existence and Uniqueness of Solutions. Chapter 6. Second and Higher Order Linear Differential Equations and Systems6.1 Homogeneous Linear Constant Coefficient Second Order Equations; 6.2 Oscillatory Solutions; 6.3 Homogeneous Linear Higher Order Constant Coefficient Equations; 6.4 Undetermined Coefficients Particular Integrals; 6.5 Cauchy-Euler Equation; 6.6 Variation of Parameters and the Green's Function; 6.7 Finding a Second Linearly Independent Solution from a Known Solution The Reduction of Order Method; 6.8 Reduction to the Standard Form u'' + f (x)u = 0; 6.9 Systems of Ordinary Differential Equations An Introduction