- Chapter 1: First-Order Differential Equations
- Chapter 10: Systems of Linear Differential Equations
- Chapter 11: Vector Differential Calculus
- Chapter 12: Vector Integral Calculus
- Chapter 13: Fourier Series
- Chapter 14: Fourier Series
- Chapter 15: Special Functions and Eigenfunction Expansions
- Chapter 16: Wave Motion on an Interval
- Chapter 17: The Heat Equation
- Chapter 18: The Potential Equation
- Chapter 19: Complex Numbers and Functions
- Chapter 2: Linear Second-Order Equations
- Chapter 20: Complex Integration
- Chapter 21: Complex Integration
- Chapter 22: The Residue Theorem
- Chapter 23: Conformal Mappings and Applications
- Chapter 3: The Laplace Transform
- Chapter 4: Series Solutions
- Chapter 5: Approximation of Solutions
- Chapter 6: Vectors and Vector Spaces
- Chapter 7: Matrices and Linear Systems
- Chapter 8: Determinants
- Chapter 9: Eigenvalues, Diagonalization, and Special Matrices
Advanced Engineering Mathematics 7th Edition - Solutions by Chapter
Full solutions for Advanced Engineering Mathematics | 7th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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!).
A sequence of steps intended to approach the desired solution.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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.
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.