 10.4.1: Suppose that the circular membrane of Example 2 has initial positio...
 10.4.2: Assume that the circular membrane of Example 2 has initial position...
 10.4.3: (a) Find u.r; t / in the case in which the circular membrane of Exa...
 10.4.4: (a) A circular plate of radius c has insulated faces and heat capac...
 10.4.5: (a) A circular plate of radius c has insulated faces and heat capac...
 10.4.6: (a) If u.r; L/ D f .r/, u.r; 0/ D 0, and the cylindrical surface r ...
 10.4.7: Let c D 1 and L D C1, so that the cylinder is semiinfinite. If u.r;...
 10.4.8: Begin with the parametric Bessel equation of order n, d dx x dy dx ...
 10.4.9: This problem provides the coefficient integrals for FourierBessel ...
 10.4.10: Suppose that f mg 1 1 are the positive roots of the equation J 0 0....
 10.4.11: If a circular membrane with fixed boundary is subjected to a period...
 10.4.12: Consider a vertically hanging cable of length L and weight w per un...
 10.4.13: Suppose that w.x/ D wx and h.x/ D h (a constant). Show that y.x; t ...
 10.4.14: Suppose that w.x/ D w (a constant) and that h.x/ D hx. Show that y....
 10.4.15: Suppose that w.x/ D wx and h.x/ D hx with w and h both constant. Sh...
 10.4.16: for the parametric Bessel equation of order zero is a regular Sturm...
 10.4.17: Suppose that an annular membrane with constant density (per unit ar...
 10.4.18: Suppose that the infinite cylindrical shell a 5 r 5 b has initial t...
Solutions for Chapter 10.4: Cylindrical Coordinate Problems
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 10.4: Cylindrical Coordinate Problems
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 and Boundary Value Problems: Computing and Modeling, edition: 5. Chapter 10.4: Cylindrical Coordinate Problems includes 18 full stepbystep solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Since 18 problems in chapter 10.4: Cylindrical Coordinate Problems have been answered, more than 15760 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

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

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

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.

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

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

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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