 5.4.1: In each of 113, find the Fourier series for the given function f o...
 5.4.2: In each of 113, find the Fourier series for the given function f o...
 5.4.3: In each of 113, find the Fourier series for the given function f o...
 5.4.4: In each of 113, find the Fourier series for the given function f o...
 5.4.5: In each of 113, find the Fourier series for the given function f o...
 5.4.6: In each of 113, find the Fourier series for the given function f o...
 5.4.7: In each of 113, find the Fourier series for the given function f o...
 5.4.8: In each of 113, find the Fourier series for the given function f o...
 5.4.9: In each of 113, find the Fourier series for the given function f o...
 5.4.10: In each of 113, find the Fourier series for the given function f o...
 5.4.11: In each of 113, find the Fourier series for the given function f o...
 5.4.12: In each of 113, find the Fourier series for the given function f o...
 5.4.13: In each of 113, find the Fourier series for the given function f o...
 5.4.14: Let f(x)=(ncosax)/2asinan, a not an integer. (a) Find the Fourier s...
 5.4.15: Suppose that f andf' are piecewise continuous on the interval  I <...
 5.4.16: Let 00 nnx f(x)=; + 2 [a,,cosT+bnsinShow that This relation is know...
 5.4.17: (a) Find the Fourier series for the function f (x)= x2 on the inter...
 5.4.18: If the Dirac delta function S (x) had a Fourier series on the inter...
 5.4.19: Derive Equations (6H8). Hint:'Use the trigonometric identities sin ...
Solutions for Chapter 5.4: Fourier series
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 5.4: Fourier series
Get Full SolutionsChapter 5.4: Fourier series includes 19 full stepbystep solutions. Since 19 problems in chapter 5.4: Fourier series have been answered, more than 6530 students have viewed full stepbystep solutions from this chapter. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Lucas numbers
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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

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(D» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.