 10.2.1: In each of 1 through 8, determine whether the given function is per...
 10.2.2: In each of 1 through 8, determine whether the given function is per...
 10.2.3: In each of 1 through 8, determine whether the given function is per...
 10.2.4: In each of 1 through 8, determine whether the given function is per...
 10.2.5: In each of 1 through 8, determine whether the given function is per...
 10.2.6: In each of 1 through 8, determine whether the given function is per...
 10.2.7: In each of 1 through 8, determine whether the given function is per...
 10.2.8: In each of 1 through 8, determine whether the given function is per...
 10.2.9: If f(x) = x for L < x < L, and if f(x + 2L) = f(x), find a formula ...
 10.2.10: If f(x) =x + 1, 1 < x < 0,x, 0 < x < 1, and if f(x + 2) = f(x), fin...
 10.2.11: If f(x) = L x for 0 < x < 2L, and if f(x + 2L) = f(x), find a formu...
 10.2.12: Verify Eqs. (6) and (7) in this section by direct integration.
 10.2.13: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.14: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.15: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.16: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.17: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.18: In each of 13 through 18:(a) Sketch the graph of the given function...
 10.2.19: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.20: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.21: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.22: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.23: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.24: In each of 19 through 24:(a) Sketch the graph of the given function...
 10.2.25: Consider the function f defined in 21, and let em(x) = f(x) sm(x).(...
 10.2.26: Consider the function f defined in 24, and let em(x) = f(x) sm(x).(...
 10.2.27: Suppose that g is an integrable periodic function with period T.(a)...
 10.2.28: If f is differentiable and is periodic with period T, show that fis...
 10.2.29: In this problem we indicate certain similarities between threedime...
Solutions for Chapter 10.2: Fourier Series
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 10.2: Fourier Series
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Chapter 10.2: Fourier Series includes 29 full stepbystep solutions. Since 29 problems in chapter 10.2: Fourier Series have been answered, more than 17961 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of 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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

GramSchmidt 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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Pseudoinverse A+ (MoorePenrose 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).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.