- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
peA) = det(A - AI) has peA) = zero matrix.
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.)
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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 symmetric matrix with eigenvalues of both signs (+ and - ).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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