 20.2.1: In 16, determine where the given complex mapping is conformal.
 20.2.2: In 16, determine where the given complex mapping is conformal.
 20.2.3: In 16, determine where the given complex mapping is conformal.
 20.2.4: In 16, determine where the given complex mapping is conformal.
 20.2.5: In 16, determine where the given complex mapping is conformal.
 20.2.6: In 16, determine where the given complex mapping is conformal.
 20.2.7: In 710, use the results in Examples 2 and 3.
 20.2.8: In 710, use the results in Examples 2 and 3.
 20.2.9: In 710, use the results in Examples 2 and 3.
 20.2.10: In 710, use the results in Examples 2 and 3.
 20.2.11: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.12: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.13: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.14: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.15: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.16: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.17: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.18: In 1118, use the conformal mappings in Appendix IV to find a confor...
 20.2.19: In 1922, use an appropriate conformal mapping and the harmonic func...
 20.2.20: In 1922, use an appropriate conformal mapping and the harmonic func...
 20.2.21: In 1922, use an appropriate conformal mapping and the harmonic func...
 20.2.22: In 1922, use an appropriate conformal mapping and the harmonic func...
 20.2.23: In 2326, use an appropriate conformal mapping and the harmonic func...
 20.2.24: In 2326, use an appropriate conformal mapping and the harmonic func...
 20.2.25: In 2326, use an appropriate conformal mapping and the harmonic func...
 20.2.26: In 2326, use an appropriate conformal mapping and the harmonic func...
 20.2.27: A realvalued function f(x, y) is called biharmonic in a domain D w...
Solutions for Chapter 20.2: Conformal Mappings
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 20.2: Conformal Mappings
Get Full SolutionsSince 27 problems in chapter 20.2: Conformal Mappings have been answered, more than 36166 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Chapter 20.2: Conformal Mappings includes 27 full stepbystep solutions.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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).

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·