 13.3.13.1.39: Solve the BVP in Example 1 if Write out the first four nonzero term...
 13.3.13.1.40: The solution u(r, u) in Example 1 of this section could also be int...
 13.3.13.1.41: Find the solution of the problem in Example 1 if f(u) cos u, 0 u p....
 13.3.13.1.42: Find the solution of the problem in Example 1 if f(u) 1 cos 2u, 0 u...
 13.3.13.1.43: Find the steadystate temperature u(r, u) within a hollow sphere a ...
 13.3.13.1.44: The steadystate temperature in a hemisphere of radius u1 r c is de...
 13.3.13.1.45: Solve when the base of the hemisphere is insulated; that is, u /2 0...
 13.3.13.1.46: Solve for r c.
 13.3.13.1.47: The timedependent temperature within a sphere of unit radius is de...
 13.3.13.1.48: A uniform solid sphere of radius 1 at an initial constant temperatu...
 13.3.13.1.49: Solve the boundaryvalue problem involving spherical vibrations: [H...
 13.3.13.1.50: A conducting sphere of radius r c is grounded and placed in a unifo...
 13.3.13.1.51: In spherical coordinates, the 3dimensional form of Helmholtzs part...
Solutions for Chapter 13.3: BoundaryValue Problems in Other Coordinate Systems
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 13.3: BoundaryValue Problems in Other Coordinate Systems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 13.3: BoundaryValue Problems in Other Coordinate Systems includes 13 full stepbystep solutions. Since 13 problems in chapter 13.3: BoundaryValue Problems in Other Coordinate Systems have been answered, more than 21239 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

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.

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!

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.