- 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
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A = CTC = (L.J]))(L.J]))T for positive definite A.
Remove row i and column j; multiply the determinant by (-I)i + j •
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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 directed graph that has constants Cl, ... , Cm associated with the edges.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.