- 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
Upper triangular systems are solved in reverse order Xn to Xl.
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!
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.)
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
Outer product uv T
= column times row = rank one matrix.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
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.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.