- 220.127.116.11.31: CAS PROJECT. Isotherms. Graph isothenns and lines of heat flow in E...
- 18.104.22.168.32: Find the temperature and the complex potential in an infinite plate...
- 22.214.171.124.33: Find the temperature between two parallel plates \" = 0 and )" = d ...
- 126.96.36.199.34: Find the temperature T in the sector 0 ~ Arg z ~ w/3, Izl ~ 1 if T ...
- 188.8.131.52.35: Find the temperature in Fig. 405 if T = -20DC on the y-axis, T = 10...
- 184.108.40.206.36: Interpret Prob. 10 in Sec. 18.2 as a heat flow problem (with bounda...
- 220.127.116.11.37: Find the temperature and the complex potential in the first quadran...
- 18.104.22.168.38: TEAM PROJECT. Piecewise Constant Boundary Temperatures. (a) A basic...
- 22.214.171.124.39: Find the temperature T(x, y) in the given thin metal plate whose fa...
- 126.96.36.199.40: Find the temperature T(x, y) in the given thin metal plate whose fa...
- 188.8.131.52.41: Find the temperature T(x, y) in the given thin metal plate whose fa...
- 184.108.40.206.42: Find the temperature T(x, y) in the given thin metal plate whose fa...
- 220.127.116.11.43: Find the temperature T(x, y) in the given thin metal plate whose fa...
- 18.104.22.168.44: Find the temperature T(x, y) in the given thin metal plate whose fa...
Solutions for Chapter 18.3: Heat Problems
Full solutions for Advanced Engineering Mathematics | 9th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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.
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.
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].
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Gram-Schmidt 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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Solvable system Ax = b.
The right side b is in the column space of A.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).