 Chapter 17.17.2: Verify the calculations in the proof of Theorem 1.
 Chapter 17.17.3: Derive the mapping in Example 2 from (2).
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 0;;; x;;...
 Chapter 17.17.5: Consider H = ~. Find the path of the image point U' of a point::. t...
 Chapter 17.1: How did we define the angle of imersection of two oriented curves, ...
 Chapter 17.17.1: Verify all calculations in Example I.
 Chapter 17.17.2: (Composition ofLFTs) Show that substituting a linear fractional tra...
 Chapter 17.17.3: (Inverse) Find the inverse of the mapping in Example 1. Show that u...
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2  I ;;; ...
 Chapter 17.17.5: Show that the Riemann surface of w = ~ consists of 11 sheets and ha...
 Chapter 17.2: At what points is a mapping w = f(::) by an analytic function not c...
 Chapter 17.17.1: Why do the images of the curves /;;:1 = COIlsl and arg :: = COllst ...
 Chapter 17.17.2: (Matrices) If you are familiar with 2 X 2 matrices, prove that the ...
 Chapter 17.17.3: Verify the formula (3) for disks.
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 0.5 < x...
 Chapter 17.17.5: Make a sketch. similar to Fig. 392, of the Riemann surface of ~.
 Chapter 17.3: What happens to angles at::o under a mapping w = f(:) if J' (Zo) = ...
 Chapter 17.17.1: Doe, the mapping w = Z = x  iy preserve angles in size as well as ...
 Chapter 17.17.2: Find the inverse;: = ;:(w). Check the result by solving ;:(w) for II".
 Chapter 17.17.3: Derive the mapping in Example 4 from (2). Find its inverse and prov...
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 3 < x <...
 Chapter 17.17.5: Show that the Riemann surtace of II = Y(::  1)(:  2) has branch p...
 Chapter 17.4: What do "surjective." "injective." and "'bijective" mean?
 Chapter 17.17.1: Find and sketch or graph the image of the given curves under the gi...
 Chapter 17.17.2: Find the inverse;: = ;:(w). Check the result by solving ;:(w) for II".
 Chapter 17.17.3: (Inverse) If w = f(z) is any transformation that has an inverse, pr...
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 0 < x < ...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.5: What mapping gave the 10ukowski airfoil?
 Chapter 17.17.1: Find and sketch or graph the image of the given curves under the gi...
 Chapter 17.17.2: Find the inverse;: = ;:(w). Check the result by solving ;:(w) for II".
 Chapter 17.17.3: CAS EXPERIMENT. Linear Fractional Transformations (LFTs). (a) Graph...
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 x < 0, ...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.6: What are linear fractional transformations (LFTs)? Why are they imp...
 Chapter 17.17.1: Find and sketch or graph the image of the given curves under the gi...
 Chapter 17.17.2: Find the inverse;: = ;:(w). Check the result by solving ;:(w) for II".
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch the image of the given region under w = e2 x arbitr...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.7: Why did we require that ad  be * 0 for a LFT?
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: CAS EXPERIMENT. Conformal Mapping. If your CAS can do conformal map...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.8: What are fixed points of a mapping? Give examples.
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = si...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.9: Can you remember mapping properties of II = sin::.? cos:? e Z ?
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = si...
 Chapter 17.17.5: Find the branch points and the number of sheets of the Riemann surf...
 Chapter 17.10: What is a Riemann surface? Why was it imroduced? Explain the simple...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = si...
 Chapter 17.11: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = si...
 Chapter 17.12: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Determine all points at which H" = sin Z IS not conformal.
 Chapter 17.13: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find the fixed points.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find and sketch or graph the images of the lines x = O. 71'/6, 71'/...
 Chapter 17.14: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: Find and sketch or graph the image of the given region under the gi...
 Chapter 17.17.2: Find a LFT whose (only) fixed points are 2 and 2.
 Chapter 17.17.3: Find the LFT that maps the given three points onto the three given ...
 Chapter 17.17.4: Find an analytic function that maps the region R bounded by the pos...
 Chapter 17.15: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: CAS EXPERI:\IENT. Orthogonal Nets. Graph the orthogonal net of the ...
 Chapter 17.17.2: Find a LFT (not w = z) with fixed points 0 and l
 Chapter 17.17.3: Find all LFTs w(:.':) that map the xaxis onto the Iaxis.
 Chapter 17.17.4: Describe the mapping H" = cosh z in terms of the mapping w = sin z ...
 Chapter 17.16: Find and sketch the image of the given curve or region under 1V = Z...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.2: Find all LFfs with fixed points I and 1.
 Chapter 17.17.3: Find a LFT that maps 1:.0:1 ~ I onto Iwl ~ 1 so that z = i/2 is map...
 Chapter 17.17.4: Find all points at which the mapping w = cosh 71;;: is not conformal.
 Chapter 17.17: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.2: Find all LFfs whose only fixed point is O.
 Chapter 17.17.3: Find an analytic function that maps the second quadrant of the zpl...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = co...
 Chapter 17.18: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.2: Find all LFfs with fixed points 0 and 00.
 Chapter 17.17.3: Find an analytic function w = i(z) that maps the region o ~ arg z ~...
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = co...
 Chapter 17.19: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.2: Find all LFfs without fixed points in the finite plane.
 Chapter 17.17.3: (Composite) Show that the composite of two LFrs is a LFT.
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = co...
 Chapter 17.20: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = co...
 Chapter 17.21: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.4: Find and sketch or graph the image of the given region under w = co...
 Chapter 17.22: Find and sketch the image of the gi ven curve or region under w = I...
 Chapter 17.17.1: Find all points at which the following mappings are not conformal.
 Chapter 17.17.4: Find the images of the lines \' = C = COllst under the mapping w = ...
 Chapter 17.23: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Find the magnification ratio M. Describe what it tell, you about th...
 Chapter 17.17.4: Show that w = Ln  maps the upper halfplane z + I onto the horizo...
 Chapter 17.24: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Find the magnification ratio M. Describe what it tell, you about th...
 Chapter 17.17.4: Find and sketch the image of R: 2 ;;; Izl ~ 3, 71'/4 ~ () ~ 71'12 u...
 Chapter 17.25: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Find the magnification ratio M. Describe what it tell, you about th...
 Chapter 17.26: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Find the magnification ratio M. Describe what it tell, you about th...
 Chapter 17.27: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Find the magnification ratio M. Describe what it tell, you about th...
 Chapter 17.28: Where is the mapping by the given function not conformal? (Give rea...
 Chapter 17.17.1: Magnification of Angles. Let f(:::) be analytic at ;:0' Suppose tha...
 Chapter 17.29: Find the LFT that maps 0, 1, 2 onto 0, i, 2i, respectively
 Chapter 17.17.1: Prove the statement in Prob. 29 for general k = I. 2, .... Hint. Us...
 Chapter 17.30: Find the LFT that maps1. 1, 2 onto O. 2, 312, respectively
 Chapter 17.31: Find the LFT that maps1, 1, i onto I, I, i, respectively
 Chapter 17.32: Find the LFT that maps1, I, i onto 1  i, 2, 0, respectively
 Chapter 17.33: Find the LFT that maps0, GO. 2 onto O. 1. "". respectively
 Chapter 17.34: Find the LFT that mapsO. i, 2i onto 0, x, 2i
 Chapter 17.35: Fixed Points. Find all fixed points of
 Chapter 17.36: Fixed Points. Find all fixed points of
 Chapter 17.37: Fixed Points. Find all fixed points of
 Chapter 17.38: Fixed Points. Find all fixed points of
 Chapter 17.39: Fixed Points. Find all fixed points of
 Chapter 17.40: Fixed Points. Find all fixed points of
 Chapter 17.41: Find an analytic function II' = .f(z) that maps:The infinite strip ...
 Chapter 17.42: Find an analytic function II' = .f(z) that maps: The intelior of th...
 Chapter 17.43: Find an analytic function II' = .f(z) that maps:The region x > 0, Y...
 Chapter 17.44: Find an analytic function II' = .f(z) that maps:The semidisk Izl <...
 Chapter 17.45: Find an analytic function II' = .f(z) that maps:The sector 0 < arg ...
Solutions for Chapter Chapter 17: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 17
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter Chapter 17 includes 150 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 150 problems in chapter Chapter 17 have been answered, more than 44305 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.