 10.2.1: In each of 1 through 8 determine whether the given function is peri...
 10.2.2: In each of 1 through 8 determine whether the given function is peri...
 10.2.3: In each of 1 through 8 determine whether the given function is peri...
 10.2.4: In each of 1 through 8 determine whether the given function is peri...
 10.2.5: In each of 1 through 8 determine whether the given function is peri...
 10.2.6: In each of 1 through 8 determine whether the given function is peri...
 10.2.7: In each of 1 through 8 determine whether the given function is peri...
 10.2.8: In each of 1 through 8 determine whether the given function is peri...
 10.2.9: Iff(x) = x for L < x < L, and iff(x + 2L) = f(x), find a formula fo...
 10.2.10: If f(x) = x + 1, 1 < x < 0, x, 0 < x < 1, and if f(x + 2) = f(x), f...
 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 functio...
 10.2.14: In each of 13 through 18: (a) Sketch the graph of the given functio...
 10.2.15: In each of 13 through 18: (a) Sketch the graph of the given functio...
 10.2.16: In each of 13 through 18: (a) Sketch the graph of the given functio...
 10.2.17: In each of 13 through 18: (a) Sketch the graph of the given functio...
 10.2.18: In each of 13 through 18: (a) Sketch the graph of the given functio...
 10.2.19: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.20: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.21: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.22: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.23: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.24: In each of 19 through 24: (a) Sketch the graph of the given functio...
 10.2.25: Consider the function f defined in and let em(x) = f(x) sm(x). (a) ...
 10.2.26: Consider the function f defined in and let em(x) = f(x) sm(x). (a) ...
 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 f i...
 10.2.29: If f is differentiable and is periodic with period T, show that f i...
Solutions for Chapter 10.2: Fourier Series
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 10.2: Fourier Series
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 29 problems in chapter 10.2: Fourier Series have been answered, more than 12844 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 10.2: Fourier Series includes 29 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9.

Block matrix.
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.

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.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).