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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the fixed points of the mappingw = (3z + 2)/(z - 1)
Find the fixed points of the mappingw = z5 + 10z3 + 10z
Find the fixed points of the mappingw = (iz + 5)/(5z + i)
Find an analytic function w = f(z) that mapsThe infinite strip 0 < y < π/4 onto the upper half plane v > 0
Find an analytic function w = f(z) that mapsThe sector 0 < arg z < π/2 onto the region u < 1.
Find an analytic function w = f(z) that mapsThe region x > 0, y > 0, xy < c onto the strip 0 < v < 1.
Under what condition on the velocity vector V in (1) will F(z) in (2) be analytic?
Find and sketch the potential between two coaxial cylinders of radii r1 and r2 having potential U1 and U2, respectively.r1 = 2.5 mm r2 = 4.0 cm, U1 = 0 V, U2 = 220 V
Find the temperature between the plates y = 0 and y = d kept at 20 and 100°C, respectively. (i) Proceed directly. (ii) Use Example 1 and a suitable mapping.
Give the details of the derivation of the series (7) from the Poisson formula (5).
Why can potential problems be modeled and solved by methods of complex analysis? For what dimensions?
Guess from physics and from Fig. 416 where on the y-axis the speed is maximum. Then calculate.
Find and sketch the potential between two coaxial cylinders of radii r1 and r2 having potential U1 and U2, respectively.r1 = 10 cm, r2 = 1 m, U1 = 10 kV, U2 = -10 kV
Show that each term of (7) is a harmonic function in the disk r < R.
What is a harmonic function? A harmonic conjugate?
Find the potential Φ in the region R in the first quadrant of the z-plane bounded by the axes (having potential U1) and the hyperbola y = 1/x (having potential U2) by mapping R onto a suitable
Calculate the speed along the cylinder wall in Fig. 416, also confirming the answer to Prob. 3. Data from Prob. 3 Guess from physics and from Fig. 416 where on the y-axis the speed is maximum. Then
Find and sketch the potential between the parallel plates having potentials U1 and U2. Find the complex potential.Plates at x1 = -5 cm, x2 = 5 cm, potentials U1 = 250 V, U2 = 500 V, respectively.
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Give some examples of potential problems considered in this chapter. Make a list of corresponding functions.
Graph equipotential lines(a) In Example 1 of the text, (b) If the complex potential is F(z) = z2, iz2, ez.(c) Graph the equipotential surfaces for F(z) = Ln z as cylinders in space.
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Find and sketch the potential between the parallel plates having potentials U1 and U2. Find the complex potential.Plates at x1 = 12 cm, x2 = 24 cm, potentials U1 = 20 kV, U2 = 8 kV, respectively.
Sketch and interpret the flow with complex potential F(z) = z.
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Write a short essay on the various assumptions made in fluid flow in this chapter.
Let D: 0 ≤ x ≤ 1/2 π, 0 ≤ y ≤ 1; D* the image of D under w = sin z; and Φ* = u2 - v2. under and What is the corresponding potential Φ in D? What are its boundary values? Sketch D and D*.
What is the complex potential of an upward parallel flow of speed K > 0 in the direction of y = x? Sketch the flow.
Show that Φ = θ/π = (1/π) arctan (y/x) is harmonic in the upper half-plane and satisfies the boundary condition Φ (x, 0) = 1 if x < 0 and 0 if x > 0, and the corresponding complex
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
State the maximum modulus theorem and mean value theorems for harmonic functions.
What happens in Prob. 7 if we replace sin z by cos z = sin (z + 1/2π)?Data from Prob. 7Let D: 0 ≤ x ≤ 1/2 π, 0 ≤ y ≤ 1; D* the image of D w = sin z; and Φ* = u2 - v2. under and What is the
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
What flow do you obtain from F(z) = -iKz, K positive real?
Graph the potentials in Probs. 7 and 9 and for two other functions of your choice as surfaces over a rectangle in the xy-plane. Find the locations of the maxima and minima by inspecting these
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Find the potential and the complex potential between the plates y = x and y = x + 10 kept at 10 V and 110 V, respectively.
Obtain the flow in Example 1 from that in Prob. 11 by a suitable conformal mapping.Data from Prob. 11What flow do you obtain from F(z) = -iKz, K positive real?
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = cos z
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Do the task in Prob. 12 if U1 = 220 V and the outer cylinder is grounded, U2 = 0.Data from Prob. 12Find the potential and complex potential between the coaxial cylinders of axis 0 (hence the
At z = ±1 in Fig. 405 the tangents to the equipotential lines as shown make equal angles. Why?
By applying a suitable conformal mapping, obtain from fig. 406 the potential Φ in the sector -1/4π
Find the real and complex potentials in the sector -π/6 ≤ θ ≤ π/6 between the boundary θ = ±π/6, kept at 0 V, and the curve x3 - 3xy2 = 1, kept at 220 V.
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = sinh 2z
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Make a list of important potential functions, with applications, from memory.
Find the linear fractional transformation z = g(Z) that maps |Z| ≤ 1 onto |z| ≤ 1 with Z = i/2 being mapped onto z = 0. Show that Z1 = 0.6 + 0.8i is mapped onto z = -1 and Z2 = -0.6 + 0.8i onto z
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = 2z2 - 2
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the potential in the first quadrant of the xy-plane if the x-axis has potential 2 kV and the y-axis is grounded.
Find the complex and real potentials in the upper half-plane with boundary values 5 kV if x < 2 and 0 if x > 2 on the x-axis.
Show that the streamlines of F(z) = 1/z and circles through the origin with centers on the y- axis.
Do Φ and a harmonic conjugate Ψ in a region R have their maximum at the same point of R?
Formulate Prob. 11 in terms of electrostatics. Data from Prob. 11 Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and
Find the temperature T in the upper half-plane if, on the x-axis, T = 30°C for x > 1 and -30°C for x < 1.
Do the same task as in Prob. 19 if the boundary values on the x-axis are V0 when -α < x < α and 0 elsewhere.Data from Prob. 19Find the complex and real potentials in the upper half-plane with
Find the streamlines and the velocity for the complex potential F(z) = (1 + i)z. Describe the flow.
Show that the isotherms of F(z) = -iz2 + z are hyperbolas.
Find V in Prob. 22 and verify that it gives vectors tangent to the streamlines.Data from Prob. 22Describe the streamlines for F(z) = 1/2z2 + z.
In your own words, and using as few formulas as possible, write a short report on spline interpolation, its motivation, a comparison with polynomial interpolation, and its applications.
Write 84.175, -528.685, 0.000924138, and -362005 in floating-point form, rounded to 5S (5 significant digits).
What is a numeric method? How has the computer influenced numerics?
What is an error? A relative error? An error bound?
Small differences of large numbers may be particularly strongly affected by rounding errors. Illustrate this by computing 0.81534/ (35 · 724 - 35.596) as given with 5S, then rounding step wise to
Solve by fixed-point iteration and answer related questions where indicated. Show details.f = x - 0.5 cos x = 0, x0 = 1. Sketch a figure.
Calculate the Lagrange polynomial P2(x) for the values Г(1.00) = 1.0000, Г(1.02) = 0.9888, Г(1.04) = 0.9784 of the gamma function [(24) in App. A3.1] and from it approximations of Г(1.01) and
Why are roundoff errors important? State the rounding rules.
What is an algorithm? Which of its properties are important in software implementation?
What do you know about stability?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Sketch f(x) = x3 - 5.00x2 + 1.01x + 1.88, showing roots near ±1 and 5. Write x = g(x) = (5.00x2 - 1.01x +
Let α1, · · ·, αn be numbers with αj correctly rounded to Sj digits. In calculating the sum α1 + · · · + αn retaining S = min Sj significant digits, is it essential that we
Find e-0.25 and e-0.75 by linear interpolation of e-x with x0 = 0, x1 = 0.5 and x0 = 0.5, x1 = 1, respectively. Then find p2(x) by quadratic interpolation of e-x with x0 = 0, x1 = 0.5, x2 =
Do the tasks in Prob. 4 for f(x) = sin 1/2πx.Data from Prob. 4Integrate f(x) = x4 from 0 to 1 by (2) with h =1, h = 0.5, h = 0.25 and estimate the error for h = 0.5 and h = 0.25 by (5).
Why is the selection of a good method at least as important on a large computer as it is on a small one?
Can the Newton (–Raphson) method diverge? Is it fast? Same questions for the bisection method.
Solve by fixed-point iteration and answer related questions where indicated. Show details.Find the smallest positive solution of sin x = e-x.
Solve x2 - 30x + 1 = 0 by (4) and by (5), using 6S in the computation. Compare and comment.
Find the quadratic polynomial that agrees with sin x at x = 0, π/4, π/2 and use it for the interpolation and extrapolation of sin x at x = -π/8, π/8, 3π/8, 5π/8. Compute the errors.
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. A, 2m = 4
What is fixed-point iteration?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Solve x4 - x - 0.12 = 0 by starting from x0 = 1.
What is the advantage of Newton’s interpolation formulas over Lagrange’s?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Find the negative solution of x4 - x - 0.12 = 0.
Do the computations in Prob. 7 with 4S and 2S.Data from Prob. 7Solve x2 - 30x + 1 = 0 by (4) and by (5), using 6S in the computation. Compare and comment.
Calculate the Lagrange polynomial p2(x) for the 5S-values f(0.25) = 0.27633, f(0.5) = 0.52050, f(1.0) = 0.84270 and from p2(x) an approximation of f (0.75) (= 0.71116).
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. B, 2m = 4
What is spline interpolation? Its advantage over polynomial interpolation?
List and compare the integration methods we have discussed.
Find the cubic spline g(x) for the given data with k0 and kn as given. If we started from the piece wise linear function in Fig. 438, we would obtain g(x) in Prob. 10 as the spline satisfying g'(-2)
Solve by fixed-point iteration and answer related questions where indicated. Show details.A partial sum of the Maclaurin series of J0(x) (Sec. 5.5) is f(x) = 1 - 1/4x2 + 1/64x4 - 1/2304x6. Conclude
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. J, 2m = 4

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