 12.6.12.1.99: In 1 and 2 solve the heat equation kuxx ut, 0 x 1, t 0, subject to ...
 12.6.12.1.100: In 1 and 2 solve the heat equation kuxx ut, 0 x 1, t 0, subject to ...
 12.6.12.1.101: In 3 and 4 solve the partial differential equation (1) subject to t...
 12.6.12.1.102: In 3 and 4 solve the partial differential equation (1) subject to t...
 12.6.12.1.103: Solve the boundaryvalue problem The partial differential equation ...
 12.6.12.1.104: Solve the boundaryvalue problem The partial differential equation ...
 12.6.12.1.105: Find a steadystate solution c(x) of the boundaryvalue problem
 12.6.12.1.106: Find a steadystate solution c(x) if the rod in is semiinfinite ex...
 12.6.12.1.107: When a vibrating string is subjected to an external vertical force ...
 12.6.12.1.108: A string initially at rest on the xaxis is secured on the xaxis a...
 12.6.12.1.109: Find the steadystate temperature u(x, y) in the semiinfinite plat...
 12.6.12.1.110: The partial differential equation where h 0 is a constant, is known...
 12.6.12.1.111: In 1318 use Method 2 of this section to solve the given boundaryva...
 12.6.12.1.112: In 1318 use Method 2 of this section to solve the given boundaryva...
 12.6.12.1.113: In 1318 use Method 2 of this section to solve the given boundaryva...
 12.6.12.1.114: In 1318 use Method 2 of this section to solve the given boundaryva...
 12.6.12.1.115: In 1318 use Method 2 of this section to solve the given boundaryva...
 12.6.12.1.116: In 1318 use Method 2 of this section to solve the given boundaryva...
Solutions for Chapter 12.6: BoundaryValue Problems in Rectangular Coordinates
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 12.6: BoundaryValue Problems in Rectangular Coordinates
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 12.6: BoundaryValue Problems in Rectangular Coordinates have been answered, more than 21576 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 12.6: BoundaryValue Problems in Rectangular Coordinates includes 18 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

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

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.