 Chapter 1: Introduction to Differential Equations
 Chapter 1.1: Definitions and Terminology
 Chapter 1.2: InitialValue Problems
 Chapter 1.3: Differential Equations as Mathematical Models
 Chapter 10: Systems of Linear FirstOrder Differential Equations
 Chapter 10.1: Theory of Linear Systems
 Chapter 10.2: Homogeneous Linear Systems
 Chapter 10.3: Solution by Diagonalization
 Chapter 10.4: Nonhomogeneous Linear Systems
 Chapter 10.5: Matrix Exponential
 Chapter 11: Systems of Nonlinear Differential Equations
 Chapter 11.1: Autonomous Systems
 Chapter 11.2: Stability of Linear Systems
 Chapter 11.3: Linearization and Local Stability
 Chapter 11.4: Autonomous Systems as Mathematical Models
 Chapter 11.5: Periodic Solutions, Limit Cycles, and Global Stability
 Chapter 12: Orthogonal Functions and Fourier Series
 Chapter 12.1: Orthogonal Functions
 Chapter 12.2: Fourier Series
 Chapter 12.3: Fourier Cosine and Sine Series
 Chapter 12.4: Complex Fourier Series
 Chapter 12.5: SturmLiouville Problem
 Chapter 12.6: Bessel and Legendre Series
 Chapter 13: BoundaryValue Problems in Rectangular Coordinates
 Chapter 13.1: Separable Partial Differential Equations
 Chapter 13.2: Classical PDEs and BoundaryValue Problems
 Chapter 13.3: Heat Equation
 Chapter 13.4: Wave Equation
 Chapter 13.5: Laplaces Equation
 Chapter 13.6: Nonhomogeneous BoundaryValue Problems
 Chapter 13.7: Orthogonal Series Expansions
 Chapter 13.8: Fourier Series in Two Variables
 Chapter 14: BoundaryValue Problems in Other Coordinate Systems
 Chapter 14.1: Polar Coordinates
 Chapter 14.2: Cylindrical Coordinates
 Chapter 14.3: Spherical Coordinates
 Chapter 15: Integral Transform Method
 Chapter 15.1: Error Function
 Chapter 15.2: Applications of the Laplace Transform
 Chapter 15.3: Fourier Integral
 Chapter 15.4: Fourier Transforms
 Chapter 15.5: Fast Fourier Transform
 Chapter 16: Numerical Solutions of Partial Differential Equations
 Chapter 16.1: Laplaces Equation
 Chapter 16.2: Heat Equation
 Chapter 16.3: Wave Equation
 Chapter 17: Functions of a Complex Variable
 Chapter 17.1: Complex Numbers
 Chapter 17.2: Powers and Roots
 Chapter 17.3: Sets in the Complex Plane
 Chapter 17.4: Functions of a Complex Variable
 Chapter 17.5: CauchyRiemann Equations
 Chapter 17.6: Exponential and Logarithmic Functions
 Chapter 17.7: Trigonometric and Hyperbolic Functions
 Chapter 17.8: Inverse Trigonometric and Hyperbolic Functions
 Chapter 18: Integration in the Complex Plane
 Chapter 18.1: Contour Integrals
 Chapter 18.2: CauchyGoursat Theorem
 Chapter 18.3: Independence of the Path
 Chapter 18.4: Cauchys Integral Formulas
 Chapter 19: Series and Residues
 Chapter 19.1: Sequences and Series
 Chapter 19.2: Taylor Series
 Chapter 19.3: Laurent Series
 Chapter 19.4: Zeros and Poles
 Chapter 19.5: Residues and Residue Theorem
 Chapter 19.6: Evaluation of Real Integrals
 Chapter 2: FirstOrder Differential Equations
 Chapter 2.1: Solution Curves Without a Solution
 Chapter 2.2: Separable Equations
 Chapter 2.3: Linear Equations
 Chapter 2.4: Exact Equations
 Chapter 2.5: Solutions by Substitutions
 Chapter 2.6: A Numerical Method
 Chapter 2.7: Linear Models
 Chapter 2.8: Nonlinear Models
 Chapter 2.9: Modeling with Systems of FirstOrder DEs
 Chapter 20: Conformal Mappings
 Chapter 20.1: Complex Functions as Mappings
 Chapter 20.2: Conformal Mappings
 Chapter 20.3: Linear Fractional Transformations
 Chapter 20.4: SchwarzChristoffel Transformations
 Chapter 20.5: Poisson Integral Formulas
 Chapter 20.6: Applications
 Chapter 3: HigherOrder Differential Equations
 Chapter 3.1: Theory of Linear Equations
 Chapter 3.10: Greens Functions
 Chapter 3.11: Nonlinear Models
 Chapter 3.12: Solving Systems of Linear Equations
 Chapter 3.2: Reduction of Order
 Chapter 3.3: Homogeneous Linear Equations with Constant Coefficients
 Chapter 3.4: Undetermined Coefficients
 Chapter 3.5: Variation of Parameters
 Chapter 3.6: CauchyEuler Equations
 Chapter 3.7: Nonlinear Equations
 Chapter 3.8: Linear Models: InitialValue Problems
 Chapter 3.9: Linear Models: BoundaryValue Problems
 Chapter 4: The Laplace Transform
 Chapter 4.1: Definition of the Laplace Transform
 Chapter 4.2: The Inverse Transform and Transforms of Derivatives
 Chapter 4.3: Translation Theorems
 Chapter 4.4: Additional Operational Properties
 Chapter 4.5: The Dirac Delta Function
 Chapter 4.6: Systems of Linear Differential Equations
 Chapter 5: Series Solutions of Linear Differential Equations
 Chapter 5.1: Solutions about Ordinary Points
 Chapter 5.2: Solutions about Singular Points
 Chapter 5.3: Special Functions
 Chapter 6: Numerical Solutions of Ordinary Differential Equations
 Chapter 6.1: Euler Methods and Error Analysis
 Chapter 6.2: RungeKutta Methods
 Chapter 6.3: Multistep Methods
 Chapter 6.4: HigherOrder Equations and Systems
 Chapter 6.5: SecondOrder BoundaryValue Problems
 Chapter 7: Vectors
 Chapter 7.1: Vectors in 2Space
 Chapter 7.2: Vectors in 3Space
 Chapter 7.3: Dot Product
 Chapter 7.4: Cross Product
 Chapter 7.5: Lines and Planes in 3Space
 Chapter 7.6: Vector Spaces
 Chapter 7.7: GramSchmidt Orthogonalization Process
 Chapter 8: Matrices
 Chapter 8.1: Matrix Algebra
 Chapter 8.10: Orthogonal Matrices
 Chapter 8.11: Approximation of Eigenvalues
 Chapter 8.12: Diagonalization
 Chapter 8.13: LUFactorization
 Chapter 8.14: Cryptography
 Chapter 8.15: An ErrorCorrecting Code
 Chapter 8.16: Method of Least Squares
 Chapter 8.17: Discrete Compartmental Models
 Chapter 8.3: Rank of a Matrix
 Chapter 8.4: Determinants
 Chapter 8.5: Properties of Determinants
 Chapter 8.6: Inverse of a Matrix
 Chapter 8.7: Cramers Rule
 Chapter 8.8: The Eigenvalue Problem
 Chapter 8.9: Powers of Matrices
 Chapter 9: Vector Calculus
 Chapter 9.1: Vector Functions
 Chapter 9.10: Double Integrals
 Chapter 9.11: Double Integrals in Polar Coordinates
 Chapter 9.12: Greens Theorem
 Chapter 9.13: Surface Integrals
 Chapter 9.14: Stokes Theorem
 Chapter 9.15: Triple Integrals
 Chapter 9.16: Divergence Theorem
 Chapter 9.17: Change of Variables in Multiple Integrals
 Chapter 9.2: Motion on a Curve
 Chapter 9.3: Curvature and Components of Acceleration
 Chapter 9.4: Partial Derivatives
 Chapter 9.5: Directional Derivative
 Chapter 9.6: Tangent Planes and Normal Lines
 Chapter 9.7: Curl and Divergence
 Chapter 9.8: Line Integrals
 Chapter 9.9: Independence of the Path
 Chapter Appendix I: Derivative and Integral Formulas
 Chapter Appendix II: Gamma Function
 Chapter Appendix III: Table of Laplace Transforms
Advanced Engineering Mathematics 6th Edition  Solutions by Chapter
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Advanced Engineering Mathematics  6th Edition  Solutions by Chapter
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters: 160. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since problems from 160 chapters in Advanced Engineering Mathematics have been answered, more than 64679 students have viewed full stepbystep answer. The full stepbystep solution to problem in Advanced Engineering Mathematics were answered by , our top Math solution expert on 03/08/18, 07:27PM.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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)ยท(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).