 10.8.1: a. Find the solution u( x, y) of Laplaces equation in the rectangle...
 10.8.2: Find the solution u( x, y) of Laplaces equation in the rectangle 0 ...
 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 ...
 10.8.5: Find the solution u(r, ) of Laplaces equation urr + 1 r ur + 1 r2 u...
 10.8.6: a. Find the solution u(r, ) of Laplaces equation in the semicircula...
 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 semiinfin...
 10.8.9: Show that equation (24) has periodic solutions only if is real. Hin...
 10.8.10: Consider the problem of finding a solution u( x, y) of Laplaces equ...
 10.8.11: Find a solution u(r, ) of Laplaces equation inside the circle r = a...
 10.8.12: a. Find the solution u( x, y) of Laplaces equation in the rectangle...
 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: Show that Laplaces equation in polar coordinates is urr + 1 r ur + ...
 10.8.16: Show that Laplaces equation in cylindrical coordinates is urr + 1 r...
 10.8.17: Show that Laplaces equation in spherical coordinates is u + 2 u + 1...
 10.8.18: a. Laplaces equation in cylindrical coordinates was found in 15. Sh...
 10.8.19: 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  11th Edition
ISBN: 9781119256007
Solutions for Chapter 10.8: Laplaces Equation
Get Full SolutionsSince 19 problems in chapter 10.8: Laplaces Equation have been answered, more than 12661 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.8: Laplaces Equation includes 19 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

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).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.