- Chapter 1: First-Order 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. Half-Range 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, Two-Dimensional Wave Equation
- Chapter 12.8: Rectangular Membrane. Double Fourier Series
- Chapter 12.9: Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel 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: Cauchy-Riemann 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: Second-Order 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. (Mass-Spring System)
- Chapter 2.5: Euler-Cauchy 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: LU-Factorization. Matrix Inversion
- Chapter 20.3: Linear Systems: Solution by Iteration
- Chapter 20.4: Linear Systems: III-Conditioning. 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 QR-Factorization
- Chapter 21: Numerics for ODEs and PDEs
- Chapter 21.1: Methods for First-Order 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: Ford-Fulkerson 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. x2-Test
- 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: Constant-Coefficient 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: Sturm-Liouville Problems. Orthogonal Functions
- Chapter 5.8: Orthogonal Eigenfunction Expansions
- Chapter 6.1: Laplace Transform. Inverse Transform. Linearity. s-Shifting
- Chapter 6.2: Transforms of Derivatives and Integrals. ODEs
- Chapter 6.3: Unit Step Function. t-Shifting
- 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. Gauss-Jordan 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, Skew-Symmetric, 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 2-Space and 3-Space
- 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:
- Chapter Chapter 10:
- 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:
- Chapter Chapter 21:
- Chapter Chapter 22:
- Chapter Chapter 23:
- Chapter Chapter 24:
- Chapter Chapter 25:
- Chapter Chapter 3:
- Chapter Chapter 4:
- Chapter Chapter 5:
- Chapter Chapter 6:
- Chapter Chapter 7:
- Chapter Chapter 8:
- Chapter Chapter 9:
Advanced Engineering Mathematics 9th Edition - Solutions by Chapter
Full solutions for Advanced Engineering Mathematics | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A sequence of steps intended to approach the desired solution.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!