 13.1: Derive the continuity equation /t +J=0 from the inhomogeneous Maxwe...
 13.2: Derive the equations of electrostatics from the Maxwell equations b...
 13.3: FromB=0 it follows that there exists a vector function A such that ...
 13.4: Show that each component of E and of B satises the wave equation.
 13.5: Derive carefully the formulas (8) and (9) for the solution of Maxwe...
 13.6: Prove that (II) and (IV) follow from the solution formulas (8)(9).
 13.7: Prove that (3) follows directly from (8)(9).
 13.8: Solve the inhomogeneous Maxwell equations.
 13.1: Assuming in (5) that F = 0, and that v is a gradient (v=), which me...
 13.2: Inparticular,inasteadyow,showthatlowpressurecorrespondstohigh veloc...
 13.1: Derive (4) and (5) from (2) and (3).
 13.2: ApointmassM isattachedtoaninnitehomogeneousstringattheorigin by mea...
 13.3: Use the orthogonality of the Legendre polynomials to derive (14).
 13.4: Derive (15).
 13.5: Repeat the problem of scattering by a sphere for the case of Neuman...
 13.6: Do the problem of scattering by an innitely long cylinder with Diri...
 13.7: Solve the problem of scattering of a point source off a plane: v+k2...
 13.1: Prove properties (4), (5), and (6) of the Dirac matrices.
 13.2: Prove that the Dirac operator is a square root of the KleinGordon o...
 13.3: FortheYangMillsequations,showthattheenergyeandthemomentum pareinvar...
 13.4: Prove the gauge invariance of the YangMills equations. See Exercise...
 13.5: Use(12.3.12)inthetableofFouriertransformstocarryoutthelaststepin th...
 13.6: Fill in the details of the derivation of (20) in three dimensions.
 13.7: RederivethesolutionoftheonedimensionalKleinGordonequationby themet...
 13.8: The telegraph equation or dissipative wave equation is utt c2u+ut =...
 13.9: Solvethetelegraphequationwith=1inonedimensionasfollows.Substituting...
 13.10: Solvetheequationuxy+u =0inthequarterplane Q ={ x > 0, y > 0}withthe...
 13.11: Let A be a skewhermitian complex 22 matrix with trace=0. Show that...
Solutions for Chapter 13: PDE PROBLEMS FROM PHYSICS
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 13: PDE PROBLEMS FROM PHYSICS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 13: PDE PROBLEMS FROM PHYSICS includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. Since 28 problems in chapter 13: PDE PROBLEMS FROM PHYSICS have been answered, more than 5522 students have viewed full stepbystep solutions from this chapter. Partial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Back substitution.
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!

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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