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study help
mathematics
advanced engineering mathematics
Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions
Solve the linear system given explicitly or by its augmented matrix. Show details.
Idempotent matrix defined by A2 = A Can you find four 2 × 2 idempotent matrices?
How can you give the rank of a matrix in terms of row vectors? Of column vectors? Of determinants?
Can you add: A row and a column vector with different numbers of components? With the same number of components? Two row vectors with the same number of components but different numbers of zeros? A vector and a scalar? A vector with four components and a 2 × 2 matrix?
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
All functions y(x) = (αx + b)e-x with any constant α and b.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details. 4y + 3z = 82x - z = 23x + 2y = 5
Nilpotent matrix defined by Bm = 0 for some m. Can you find three 2 × 2 nilpotent matrices?
Showing the details, evaluate: 10.4 4.9 1.5 -1.3
Solve the linear system given explicitly or by its augmented matrix. Show details. -2y - 2z = -83x + 4y - 5z = 13
Can you prove (10a)–(10c) for 3 × 3 matrices? For m × n matrices?
What is the idea of Gauss elimination and back substitution?
Find the inverse by Gauss–Jordan (or by (4*) if n = 2). Check by using (1).
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined. 3A, 0.5B, 3A + 0.5B, 3A + 0.5B + C
All 2 x 2 matrices [αjk] with α11 + α22 = 0.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose.
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details.
What is the inverse of a matrix? When does it exist? How would you determine it?
Showing the details, evaluate:
Solve the linear system given explicitly or by its augmented matrix. Show details.
Showing all intermediate results, calculate the following expression or give reasons why they are undefined: AB, ABT, BA, BTA
Find the eigenvalues. Find the corresponding eigenvectors. Use the given λ or factor in Probs. 11 and 15.
Show that the inverse of a skew-symmetric matrix is skew-symmetric.
What do you know about directional derivatives? Their relation to the gradient?
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:a × b, b × a, a • b
Find the work done by a force p acting on a body if the body is displaced along the straight segment A̅B̅ from A to B. Sketch A̅B̅ and p. Show the details.p = [0, 4, 3], A: (4, 5, -1), B: (1, 3, 0)
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:4b × 3c, 12|b × c|, 12|c × b|
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:b × b, (b - c) × (c - b), b • b
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:b, c
Find the resultant in terms of components and its magnitude.p = [-1, 2, -3], q = [1, 1, 1], u = [1, -2, 2]
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:What will happen to the angle in Prob. 24 if we replace c by nc with larger and larger n?Data from prob. 24Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:a + c, b + c
Find the resultant in terms of components and its magnitude.u = [3, 1, -6], v = [0, 2, 5], w = [3, -1, -13]
Same question as in Prob. 34 for two ships moving northeast with speed |vA| = 22 knots and west with speed |vB| = 19 knots.Data from Prob. 34If airplanes A and B are moving southwest with speed |vA| = 550 mph, and northwest with speed |vB| = 450 mph, respectively, what is the relative velocity v =
Forces acting on moving objects (cars, airplanes, ships, etc.) require the engineer to know corresponding tangential and normal accelerations. In Probs. 35–38 find them, along with the velocity and speed. Sketch the path.r(t) = (R sin ωt + Rt)i + (R cos ωt + R)j.This is the path of a point on
The use of a CAS may greatly facilitate the investigation of more complicated paths, as they occur in gear transmissions and other constructions. To grasp the idea, using a CAS, graph the path and find velocity, speed, and tangential and normal acceleration.r(t) = [cos t + cos 2t, sin t - sin 2t]
Represent in polar form and graph in the complex plane as in Fig. 325. Do these problems very carefully because polar forms will be needed frequently. Show the details. 1 + i
Formulas for hyperbolic functionsShow thatcosh z = cosh x cos y + i sinh x sin ysinh z = sinh x cos y + i cosh x sin y
Prove analyticity of Ln z by means of the Cauchy–Riemann equations in polar form.
Find ez in the form u + iv and |ez| if z equals√2 + 1/2πi
Write in exponential form (6):4 + 3i
Using the definitions, prove:sin z1 cos z2 = 1/2 [sin (z1 + z2) + sin (z1 - z2)]
Find Re and Im ofexp (z3)
Find and graph all roots in the complex plane.4√i
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.u = xy
Apply the program in Prob. 25 to u = ex cos y, v = ex sin y and to an example of your own.Data from Prob. 25Write a program for graphing equipotential lines u = const of a harmonic function u and of its conjugate v on the same axes. Apply the program to(a) u = x2 - y2, v = 2xy(b) u = x3 - 3xy2, v =
Find and graph all roots in the complex plane.8√1
Find the principal value. Show details.(-1)2-i
Show that if u is harmonic and v is a harmonic conjugate of u, then u is a harmonic conjugate of -v.
Find and graph all roots in the complex plane.5√-1
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.v = -e-2x sin 2y
What is it? Its role? What motivates its name? How can you find it?
Illustrate Prob. 27 by an example.Data from Prob. 27Show that if u is harmonic and v is a harmonic conjugate of u, then u is a harmonic conjugate of -v.
How can you find the answer to Prob. 24 from the answer to Prob. 23?Data from Prob. 24Find the principal value. Show details.(1 - i)1+iData from Prob. 23Find the principal value. Show details.(1 + i)1-i
Formulas (4), (5), and (11) (below) are needed from time to time. Derive
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.u = exp(-(x2 - y2)/2) cos xy
Solve and graph the solutions. Show details.z2 + z + 1 - i = 0
Find the value of:cos (3 - i)
Solve and graph the solutions. Show details.z4 - 6iz2 + 16 = 0
Find the value of:tan i
Inequalities and equalityProve (6).
Find the value of:cosh (π + πi)
Inequalities and equalityProve and explain the name|z1 + z2|2 + |z1 - z2|2 = 2(|z1|2 + |z2|2).
Integrate counterclockwise around the unit circle.
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z + 1| = 1
Find the path and sketch it.z(t) = (1 + 1/2i)t (2 ≤ t ≤ 5)
What is a parametric representation of a curve? What is its advantage?
What did we assume about paths of integration z = z(t)? What is ż = dz/dt geometrically?
Integrate counterclockwise around the unit circle.
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z + i| = 1.4
Find the path and sketch it.z(t) = t + 2it2 (1 ≤ t ≤ 2)
Can we conclude from Example 4 that the integral is also zero over the contour in Prob. 1?Data from Prob. 1Verify Theorem 1 for the integral of over the boundary of the square with vertices ±1 ±i. Use deformation.
State the definition of a complex line integral from memory.
Can you remember the relationship between complex and real line integrals discussed in this chapter?
Integrate counterclockwise around the unit circle.
Integrate the given function around the unit circle.(cos 3z)/(6z)
Find the path and sketch it.z(t) = 3 - i + √10e-it (0 ≤ t ≤ 2π)
What is the connectedness of the domain in which (cos z2)/(z4 + 1) is analytic?
How can you evaluate a line integral of an analytic function? Of an arbitrary continous complex function?
What value do you get by counterclockwise integration of 1/z around the unit circle? You should remember this. It is basic.
Integrate counterclockwise around the unit circle.
Integrate the given function around the unit circle.z3/(2z - i)
Find the path and sketch it.z(t) = 2 + 4eπit/2 (0 ≤ t ≤ 2)
Can we conclude in Example 2 that the integral of 1/(z2 + 4) over(a) |z - 2| = 2(b) |z - 2| = 3 is zero?
Which theorem in this chapter do you regard as most important? State it precisely from memory.
What is independence of path? Its importance? State a basic theorem on independence of path in complex.
Integrate. Show the details. Begin by sketching the contour. Why?
Experiment to find out to what extent your CAS can do contour integration. For this, use(a) The second method in Sec. 14.1.(b) Cauchy’s integral formula.
Find the path and sketch it.z(t) = t + it3 (-2 ≤ t ≤ 2)
What is deformation of path? Give a typical example.
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = exp (z2)
Don’t confuse Cauchy’s integral theorem (also known as Cauchy–Goursat theorem) and Cauchy’s integral formula. State both. How are they related?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.Segment from (-1, 1) to (1, 3)
What is a doubly connected domain? How can you extend Cauchy’s integral theorem to it?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(2z - 1)
What do you know about derivatives of analytic functions?
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