- 14.2.1: Find the displacement u(r, t) in Example 1 if f (r) 0 and the circu...
- 14.2.2: A circular membrane of radius 1 is clamped along its circumference....
- 14.2.3: Find the steady-state temperature u(r, z) in the cylinder in Exampl...
- 14.2.4: If the lateral side of the cylinder in Example 2 is insulated, then...
- 14.2.5: In 58, find the steady-state temperature u(r, z) in a finite cylind...
- 14.2.6: In 58, find the steady-state temperature u(r, z) in a finite cylind...
- 14.2.7: In 58, find the steady-state temperature u(r, z) in a finite cylind...
- 14.2.8: In 58, find the steady-state temperature u(r, z) in a finite cylind...
- 14.2.9: The temperature in a circular plate of radius c is determined from ...
- 14.2.10: Solve if the edge r c of the plate is in sulated.
- 14.2.11: When there is heat transfer from the lateral side of an infinite ci...
- 14.2.12: Find the steady-state temperature u(r, z) in a semi-infinite cylind...
- 14.2.13: In 13 and 14, use the substitution u(r, t) v(r, t) c(r) to solve th...
- 14.2.14: In 13 and 14, use the substitution u(r, t) v(r, t) c(r) to solve th...
- 14.2.15: The horizontal displacement u(x, t) of a heavy chain of length L os...
- 14.2.16: Consider the boundary-value problem 02 u 0r2 1 r 0u 0r 0u 0t , 0 , ...
- 14.2.17: Solve the boundary-value problem 02 u 0r2 1 r 0u 0r 2 1 r2 u 02 u 0...
- 14.2.18: In this problem we consider the general casethat is, with u depende...
- 14.2.19: (a) Consider Example 1 with a 1, c 10, g(r) 0, and f (r) 1 r/10, 0 ...
- 14.2.20: Solve with boundary conditions u(c, t) 200, u(r, 0) 0. With these i...
- 14.2.21: Consider an idealized drum consisting of a thin membrane stretched ...
Solutions for Chapter 14.2: Cylindrical Coordinates
Full solutions for Advanced Engineering Mathematics | 6th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
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).
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).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
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.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.