- 10.8.1: (a) Find the solution u(x, y) of Laplaces equation in the rectangle...
- 10.8.2: (a) Find the solution u(x, y) of Laplaces equation in the rectangle...
- 10.8.3: (a) Find the solution u(x, y) of Laplaces equation in the rectangle...
- 10.8.4: Show how to find the solution u(x, y) of Laplaces equation in the r...
- 10.8.5: Find the solution u(r, ) of Laplaces equation urr + (1/r)ur + (1/r2...
- 10.8.6: (a) Find the solution u(r, ) of Laplaces equation in the semicircul...
- 10.8.7: Find the solution u(r, ) of Laplaces equation in the circular secto...
- 10.8.8: (a) Find the solution u(x, y) of Laplaces equation in the semi-infi...
- 10.8.9: Show that Eq. (23) has periodic solutions only if is real. Hint: Le...
- 10.8.10: Consider the problem of finding a solution u(x, y) of Laplaces equa...
- 10.8.11: Find a solution u(r, ) of Laplaces equation inside the circle r = a...
- 10.8.12: Find a solution u(r, ) of Laplaces equation inside the circle r = a...
- 10.8.13: (a) Find the solution u(x, y) of Laplaces equation in the rectangle...
- 10.8.14: (a) Find the solution u(x, y) of Laplaces equation in the rectangle...
- 10.8.15: By writing Laplaces equation in cylindrical coordinates r, , and z ...
- 10.8.16: Flow in an Aquifer. Consider the flow of water in a porous medium, ...
Solutions for Chapter 10.8: Laplaces Equation
Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition
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!
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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).
= Xl (column 1) + ... + xn(column n) = combination of columns.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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·