 Chapter 1: FirstOrder ODEs
 Chapter 1.1: Basic Concepts. Modeling
 Chapter 1.2: Geometric Meaning of y' = f(x, y). Direction Fields
 Chapter 1.3: Separable ODEs. Modeling
 Chapter 1.4: Exact ODEs. Integrating Factors
 Chapter 1.5: Linear ODEs. Bernoulli Equation. Population Dynamics
 Chapter 1.6: Orthogonal Trajectories. Optional
 Chapter 1.7: Existence and Uniqueness of Solutions
 Chapter 10: Vector Integral Calculus. Integral Theorems
 Chapter 10.1: Line Integrals
 Chapter 10.2: Path Independence of Line Integrals
 Chapter 10.3: Calculus Review: Double Integrals. Optional
 Chapter 10.4: Green's Theorem in the Plane
 Chapter 10.5: Surfaces for Surface Integrals
 Chapter 10.6: Surface Integrals
 Chapter 10.7: Triple Integrals. Divergence Theorem of Gauss
 Chapter 10.8: Further Applications of the Divergence Theorem
 Chapter 10.9: Stokes's Theorem
 Chapter 11.1: Fourier Series
 Chapter 11.10: Tables of Transforms
 Chapter 11.2: Functions of Any Period p = 2L
 Chapter 11.3: Even and Odd Functions. HalfRange Expansions
 Chapter 11.4: Complex Fourier Series. Optiollal
 Chapter 11.5: Forced Oscillations
 Chapter 11.6: Approximation by Trigonometric Polynomials
 Chapter 11.7: Fourier Integral
 Chapter 11.8: Fourier Cosine and Sine Transforms
 Chapter 11.9: Fourier Transform. Discrete and Fast Fourier Transforms
 Chapter 12: Fourier Series, Integrals, and Transforms
 Chapter 12.1: Basic Concepts
 Chapter 12.10: Laplace's Equation in Cylindrical and Spherical Coordinates. Potential
 Chapter 12.11: Solution of PDEs by Laplace Transforms
 Chapter 12.2: Modeling: Vibrating String, Wave Equation
 Chapter 12.3: Solution by Separating Vatiables. Use of Fourier Series
 Chapter 12.4: D' Alembert's Solution of the Wave Equation. Characteristics
 Chapter 12.5: Heat Equation: Solution by Fourier Series
 Chapter 12.6: Heat Equation: Solution by Fourier Integrals and Transforms
 Chapter 12.7: Modeling: Membrane, TwoDimensional Wave Equation
 Chapter 12.8: Rectangular Membrane. Double Fourier Series
 Chapter 12.9: Laplacian in Polar Coordinates. Circular Membrane. FourierBessel Series
 Chapter 13: Complex Analysis
 Chapter 13.1: Complex Numbers. Complex Plane
 Chapter 13.2: Polar Form of Complex Numbers. Powers and Roots
 Chapter 13.3: Derivative. Analytic Function
 Chapter 13.4: CauchyRiemann Equations. Laplace's Equation
 Chapter 13.5: Exponential Function
 Chapter 13.6: Trigonometric and Hyperbolic Functions
 Chapter 13.7: Logarithm. General Power
 Chapter 14: Complex Integration
 Chapter 14.1: Line Integral in the Complex Plane
 Chapter 14.2: Cauchy's Integral Theorem
 Chapter 14.3: Cauchy's Integral Formula
 Chapter 14.4: Derivatives of Analytic Functions
 Chapter 15: Power Series, Taylor Series
 Chapter 15.1: Sequences, Series, Convergence Tests
 Chapter 15.2: Power Series
 Chapter 15.3: Functions Given by Power Series
 Chapter 15.4: Taylor and Maclaurin Series
 Chapter 15.5: Uniform Convergence. Optional
 Chapter 16: Laurent Series. Residue Integration
 Chapter 16.1: Laurent Series
 Chapter 16.2: Singularities and Zeros. Infinity
 Chapter 16.3: Residue Integration Method
 Chapter 16.4: Residue Integration of Real Integrals
 Chapter 17: Conformal Mapping
 Chapter 17.1: Geometry of Analytic Functions: Conformal Mapping
 Chapter 17.2: Linear Fractional Transformations
 Chapter 17.3: Special Linear Fractional Transformations
 Chapter 17.4: Conformal Mapping by Other Functions
 Chapter 17.5: Riemann Surfaces. Optional
 Chapter 18: Complex Analysis and Potential Theory
 Chapter 18.1: Electrostatic Fields
 Chapter 18.2: Use of Conformal Mapping. Modeling
 Chapter 18.3: Heat Problems
 Chapter 18.4: Ruid Flow
 Chapter 18.5: Poisson's Integral Formula for Potentials
 Chapter 18.6: General Properties of Harmonic Function..
 Chapter 19: Numerics in General
 Chapter 19.1: Introduction
 Chapter 19.2: Solution of Equations by Iteration
 Chapter 19.3: [nterpolation
 Chapter 19.4: Spline Interpolation
 Chapter 19.5: Numeric Integration and Differentiation
 Chapter 2: SecondOrder Linear ODEs
 Chapter 2.1: Homogeneous Linear ODEs of Second Order
 Chapter 2.10: Solution by Variation of Parameters
 Chapter 2.2: Homogeneous Linear ODEs with Constant Coefficients
 Chapter 2.3: Differential Operators. Optional
 Chapter 2.4: Modeling: Free Oscillations. (MassSpring System)
 Chapter 2.5: EulerCauchy Equations
 Chapter 2.6: Existence and Uniqueness of Solutions. Wronskian
 Chapter 2.7: Nonhomogeneous ODEs
 Chapter 2.8: Modeling: Forced Oscillations. Resonance
 Chapter 2.9: Modeling: Electric Circuits
 Chapter 20: Numeric Linear Algebra
 Chapter 20.1: Linear Systems: Gauss EliminatIOn
 Chapter 20.2: Linear Systems: LUFactorization. Matrix Inversion
 Chapter 20.3: Linear Systems: Solution by Iteration
 Chapter 20.4: Linear Systems: IIIConditioning. Norms
 Chapter 20.5: Least Squares Method
 Chapter 20.6: Matrix Eigenvalue Problems: Introduction
 Chapter 20.7: [ncIusion of Matrix Eigenvalues
 Chapter 20.8: Power Method for Eigenvalues
 Chapter 20.9: Tridiagonalization and QRFactorization
 Chapter 21: Numerics for ODEs and PDEs
 Chapter 21.1: Methods for FirstOrder ODEs
 Chapter 21.2: Multistep Methods
 Chapter 21.3: Methods for Systems and Higher Order ODEs
 Chapter 21.4: Methods for Elliptic PDEs
 Chapter 21.5: Neumann and Mixed Problems. Inegular Boundary
 Chapter 21.6: Methods for Parabolic PDEs
 Chapter 21.7: Method for Hyperbolic PDEs
 Chapter 22: Unconstrained Optimization. Linear Programming
 Chapter 22.1: Basic Concepts. Unconstrained Optimization
 Chapter 22.2: Linear Programming
 Chapter 22.3: Simplex Method
 Chapter 22.4: Simplex Method: Difficulties
 Chapter 23: Combinatorial Optimization
 Chapter 23.1: Graphs and Digraphs
 Chapter 23.2: Shortest Path Problems. Complexity
 Chapter 23.3: Bellman's Principle. Dijkstra's Algorithm
 Chapter 23.4: Shortest Spanning Trees. Greedy Algorithm
 Chapter 23.5: Shortest Spanning Trees. Prim's Algorithm
 Chapter 23.6: Flows in Networks
 Chapter 23.7: Maximum Flow: FordFulkerson Algorithm
 Chapter 23.8: Bipartite Graphs. Assignment Problem~
 Chapter 24: Data Analysis. Probability Theory
 Chapter 24.1: Data Representation. Average. Spread
 Chapter 24.2: Experiments, Outcomes, Events
 Chapter 24.3: Probability
 Chapter 24.4: Permutations and Combinations
 Chapter 24.5: Random Variables. Probability Distributions
 Chapter 24.6: Mean and Variance of a Distribution
 Chapter 24.7: Binomial. Poisson, and Hypergeometric Distributions
 Chapter 24.8: Normal Distribution
 Chapter 24.9: Distributions of Several Random Variables
 Chapter 25: Mathematical Statistics
 Chapter 25.2: Point Estimation of Parameters
 Chapter 25.3: Confidence Intervals
 Chapter 25.4: Testing Hypotheses. Decisions
 Chapter 25.5: Quality Control
 Chapter 25.6: Acceptance Sampling
 Chapter 25.7: Goodness of Fit. x2Test
 Chapter 25.8: Nonparametric Tests
 Chapter 25.9: Regression. Fitting Straight Lines. Correlation
 Chapter 3.1: Homogeneous Linear ODEs
 Chapter 3.2: Homogeneous Linear ODEs with Constant Coefficients
 Chapter 3.3: Nonhomogeneous Linear ODEs
 Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods
 Chapter 4.1: Systems of ODEs as Models
 Chapter 4.3: ConstantCoefficient Systems. Phase Plane Method
 Chapter 4.4: Criteria for Critical Points. Stability
 Chapter 4.5: Qualitative Methods for Nonlinear Systems
 Chapter 4.6: Nonhomogeneous Linear Systems of ODEs
 Chapter 5: Series Solutions of ODEs. Special Functions
 Chapter 5.1: Power Series Method
 Chapter 5.2: Theory of the Power Series Method
 Chapter 5.3: Legendre's Equation. Legendre Polynomials P nex)
 Chapter 5.4: Frobenius Method
 Chapter 5.5: Bessel's Equation. Bessel Functions lvCx)
 Chapter 5.6: Bessel Functions of the Second Kind YvCx)
 Chapter 5.7: SturmLiouville Problems. Orthogonal Functions
 Chapter 5.8: Orthogonal Eigenfunction Expansions
 Chapter 6.1: Laplace Transform. Inverse Transform. Linearity. sShifting
 Chapter 6.2: Transforms of Derivatives and Integrals. ODEs
 Chapter 6.3: Unit Step Function. tShifting
 Chapter 6.4: Short Impulses. Dirac's Delta Function. Pm1ial Fractions
 Chapter 6.5: Convolution. Integral Equations
 Chapter 6.6: Differentiation and Integration of Transforms.
 Chapter 6.7: Systems of ODEs
 Chapter 6.8: Laplace Transform: General Formulas
 Chapter 7: Linear Algebra. Vector Calculus
 Chapter 7.1: Matrices, Vectors: Addition and Scalar Multiplication
 Chapter 7.2: Matrix Multiplication
 Chapter 7.3: Linear Systems of Equations. Gauss Elimination
 Chapter 7.4: Linear Independence. Rank of a Matrix. Vector Space
 Chapter 7.7: Determinants. Cramer's Rule
 Chapter 7.8: Inverse of a Matrix. GaussJordan Elimination
 Chapter 7.9: Vector Spaces, Inner Product Spaces. Linear Transformations. Optional
 Chapter 8: Linear Algebra: Matrix Eigenvalue Problems
 Chapter 8.1: Eigenvalues, Eigenvectors
 Chapter 8.2: Some Applications of Eigenvalue Problems
 Chapter 8.3: Symmetric, SkewSymmetric, and Orthogonal Matrices
 Chapter 8.4: Eigenbases. Diagonalization. Quadratic Forms
 Chapter 8.5: Complex Matrices and Forms. Optional
 Chapter 9: Vector Differential Calculus. Grad, Div, Curl
 Chapter 9.1: Vectors in 2Space and 3Space
 Chapter 9.2: Inner Product (Dot Product)
 Chapter 9.3: Vector Product (Cross Product)
 Chapter 9.4: Vector and Scalar Functions and Fields. Derivatives
 Chapter 9.5: Curves. Arc Length. Curvature. Torsion
 Chapter 9.6: Calculus Review: Functions of Several Variables. Optional
 Chapter 9.7: Gradient of a Scalar Field. Directional Derivative
 Chapter 9.8: Divergence of a Vector Field
 Chapter 9.9: Curl of a Vector Field
 Chapter CHAPTER 1:
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 Chapter Chapter 11:
 Chapter Chapter 12:
 Chapter Chapter 13:
 Chapter Chapter 14:
 Chapter Chapter 15:
 Chapter Chapter 16:
 Chapter Chapter 17:
 Chapter Chapter 18:
 Chapter Chapter 19:
 Chapter CHAPTER 2:
 Chapter Chapter 20:
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 Chapter Chapter 22:
 Chapter Chapter 23:
 Chapter Chapter 24:
 Chapter Chapter 25:
 Chapter Chapter 3:
 Chapter Chapter 4:
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Advanced Engineering Mathematics 9th Edition  Solutions by Chapter
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ISBN: 9780471488859
Advanced Engineering Mathematics  9th Edition  Solutions by Chapter
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Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Outer product uv T
= column times row = rank one matrix.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.