- Chapter 1: Introduction to Differential Equations
- Chapter 1.1: Definitions and Terminology
- Chapter 1.2: Initial-Value Problems
- Chapter 1.3: Differential Equations as Mathematical Models
- Chapter 10: Systems of Linear First-Order 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: Boundary-Value Problems in Rectangular Coordinates
- Chapter 13.1: Separable Partial Differential Equations
- Chapter 13.2: Classical PDEs and Boundary-Value Problems
- Chapter 13.3: Heat Equation
- Chapter 13.4: Wave Equation
- Chapter 13.5: Laplaces Equation
- Chapter 13.6: Nonhomogeneous Boundary-Value Problems
- Chapter 13.7: Orthogonal Series Expansions
- Chapter 13.8: Fourier Series in Two Variables
- Chapter 14: Boundary-Value 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: First-Order 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 First-Order 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: Higher-Order 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: Initial-Value Problems
- Chapter 3.9: Linear Models: Boundary-Value 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: Higher-Order Equations and Systems
- Chapter 6.5: Second-Order Boundary-Value Problems
- Chapter 7: Vectors
- Chapter 7.1: Vectors in 2-Space
- Chapter 7.2: Vectors in 3-Space
- Chapter 7.3: Dot Product
- Chapter 7.4: Cross Product
- Chapter 7.5: Lines and Planes in 3-Space
- 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: LU-Factorization
- Chapter 8.14: Cryptography
- Chapter 8.15: An Error-Correcting 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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.
Every v in V is orthogonal to every w in W.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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