 14.1.1: In 14, find the steadystate temperature u(r, u) in a circular plat...
 14.1.2: In 14, find the steadystate temperature u(r, u) in a circular plat...
 14.1.3: In 14, find the steadystate temperature u(r, u) in a circular plat...
 14.1.4: In 14, find the steadystate temperature u(r, u) in a circular plat...
 14.1.5: If the boundaries u 0 and u p of a semicircular plate of radius 2 a...
 14.1.6: Find the steadystate temperature u(r, u) in a semicircular plate o...
 14.1.7: Find the steadystate temperature u(r, u) in the quartercircular p...
 14.1.8: Find the steadystate temperature u(r, u) in the quartercircular p...
 14.1.9: Find the steadystate temperature u(r, u) in the portion of a circu...
 14.1.10: Find the steadystate temperature u(r, u) in the infinite wedgeshap...
 14.1.11: Find the steadystate temperature u(r, u) in the plate in the form ...
 14.1.12: If the boundaryconditions for the annular plate in Figure 14.1.7 a...
 14.1.13: Find the steadystate temperature u(r, u) in the annular plate show...
 14.1.14: Find the steadystate temperature u(r, u) in the annular plate show...
 14.1.15: Find the steadystate temperature u(r, u) in the semiannular plate ...
 14.1.16: Find the steadystate temperature u(r, u) in the semiannular plate ...
 14.1.17: Find the steadystate temperature u(r, u) in the quarterannular pl...
 14.1.18: The plate in the first quadrant shown in FIGURE 14.1.10 is oneeigh...
 14.1.19: Solve the exterior Dirichlet problem for a circular disk of radius ...
 14.1.20: Consider the steadystate temperature u(r, u) in the semiannular pl...
 14.1.21: (a) Find the series solution for u(r, u) in Example 1 when u(1, u) ...
 14.1.22: Solve the Neumann problem for a circular plate: 02 u 0r2 1 r 0u 0r ...
 14.1.23: Consider the annular plate shown in Figure 14.1.7. Discuss how the ...
 14.1.24: Verify that u(r, u) 3 4 r sinu 2 1 4r3 sin3u is a solution of the b...
Solutions for Chapter 14.1: Polar Coordinates
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 14.1: Polar Coordinates
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 14.1: Polar Coordinates includes 24 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Since 24 problems in chapter 14.1: Polar Coordinates have been answered, more than 39176 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Lucas numbers
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.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.