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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the potential and complex potential between the coaxial cylinders of axis 0 (hence the vertical axis in space) and radii r1 = 1 cm, r2 = 10 cm, kept at potential U1 = 220 V and U2 = 2 kV,
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
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
(a) Verify Theorem 3 for(i) F(z) = z2 and the rectangle 1 ≤ x ≤ 5, 2 ≤ y ≤ 4(ii) F(z) = sin z and the unit disk(iii) F(z) = ez and any bounded domain.(b) F(z) = 1 + |z| is not zero in the
Find and sketch the potential between the axes (potential 500 V) and the hyperbola xy = 4 (potential 100 V).
Find the potential between the cylinders C1: |z| = 1 (potential U1 = 0) and C2: |z - c| = c (potential U2 = 220 V), where 0 < c < 1/2. Sketch or graph equipotential lines and their orthogonal
State Poisson’s integral formula. Derive it from Cauchy’s formula.
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
What happens in Prob. 7 if you replace the potential by its conjugate harmonic?Data from Prob. 7Let D: 0 ≤ x ≤ 1/2 π, 0 ≤ y ≤ 1; D* the image of D under w = sin z; and Φ* = u2 - v2. What is
Explain the use of conformal mapping in potential theory.
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
Verify (3) in Theorem 2 for the given Φ(x, y), (x0, y0) and circle of radius 1.x2 - y2, (3, 8)
Apply Theorem 1 to Φ*(u, v) = u2 - v2, w = f(z) = ez, and any domain D, showing that the resulting potential Φ is harmonic.
What does the complex potential give physically?
Derive the first statement in Theorem 2 from Poisson’s integral formula.
Sketch or graph and interpret the flow in Example 1 on the whole upper half-plane.
Find and sketch the potential between the parallel plates having potentials U1 and U2. Find the complex potential.Plates at y = x and y = x + k potentials U1 = 0 V, U2 = 220 V, respectively.
Let Φ* = 4uv, w = f(z) = ez, and D: x < 0, 0 < y < π. Find Φ. What are its boundary values?
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
Verify Theorem 1 for the given F(z), z0, and circle of radius 1.(z - 1)-2, z0 = -1
Find and sketch the potential between two coaxial cylinders of radii r1 and r2 having potential U1 and U2, respectively.If r1 = 2 cm, r2 = 6 cm and U1 = 300 V, U2 = 100 V, respectively, is the
Give the details of the steps. What is the point of that proof?
What parts of complex analysis are mainly of interest to the engineer and physicist?
Find the temperature and the complex potential in an infinite plate with edges y = x - 4 and y = x + 4 kept at -20 and 40°C, respectively (Fig. 413). In what case will this be an approximate
Verify Theorem 1 for the given F(z), z0, and circle of radius 1.2z4, z0 = -2
Find and sketch the potential between two coaxial cylinders of radii r1 and r2 having potential U1 and U2, respectively.r1 = 1 mm r2 = 2 cm, U1 = 400 V, U2 = 0 V
Along what curves will the speed in Example 1 be constant? Is this obvious from Fig. 415? の
Find the LFT that maps0, 1, 2 onto 2i, 1 + 2i, 2 + 2i, respectively
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = z3
Find all points at which the mapping is not conformal. Give reason.z2 + 1/z2
Evaluate (counterclockwise). Show the details. C:|z – il = 3 dz ,2 (z? + 1)3
Find all Taylor and Laurent series with center z0. Determine the precise regions of convergence. Show details.1/z, z0 = 1
Find all LFTs with fixed point(s).z = ±1
Determine the location and order of the zeros.(sin z - 1)3
Find all the singularities in the finite plane and the corresponding residues. Show the details.cos z/z4
Is the given series convergent or divergent? Give a reason. Show details. Vn п-1
Find the radius of convergence in two ways:(a) Directly by the Cauchy€“Hadamard formula.(b) From a series of simpler terms by using Theorem 3 or Theorem 4. Σ( п+ m п т п-0
Find the radius of convergence. -(z – 3i)2n n! n-1
Solve for z.ln z = e - πi
Find all solutions.cosh z = -1
Represent in polar form, with the principal argument.12 + i, 12 - i
Graph in the complex plane and represent in the form x + iy:√50(cos 3/4π + i sin 3/4π)
Find the value of the derivative of(z - i)/(z + i) at i
Write in exponential form (6):n√z
Find the potential in the following charge-free regions.Between two concentric spheres of radii r0 and r1 kept at potentials u0 and u1, respectively.
This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x,
Find and sketch or graph the deflection u(x, t) of a vibrating string of length Ï€, extending from x = 0 to x = Ï€, and c2= T/Ï = 4 starting with velocity zero and
By the principles used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 292) are modeled by the fourth-order PDEwhere c2 = EI/ÏA
Solve for u = u(x, y):uxx + ux = 0, u (0, y) = f (y), ux (0, y) = g(y)
Find the eigenvalues and eigenfunctions. Verify orthogonality. Start by writing the ODE in the form (1), using Prob. 6. Show details of your work.Orthogonal polynomials play a great role in
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work. -4 4
f(x) = |sin x| (-π < x < π), full-wave rectifier.
Obtain the solution to Prob. 26 from that of Prob. 27.Data from Prob. 26Find (a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the
Find the steady-state current I(t) in the RLC-circuit in Fig. 275, where R = 10 Î©, L = 1 H, C = 10-1 F and with E(t) V as follows and periodic with period 2Ï€. Graph or sketch
Represent f(x) as an integral (11). (cos x if f(x) 0
Represent f(x) as an integral (11). (x if 0
(a) Show that if f(x) has a Fourier transform, so does f(x - Î±), and F{f(x - Î±)} = e-iwÎ±F{f(x)}.(b) Using (a), obtain formula 1 in Table III, from formula
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.f (x) = x|x| (-1 < x < 1), p = 2
Find the steady-state oscillations of y" + cy' + y = r(t) with c > 0 and r(t) as given. The spring constant is k = 1. Show the details and sketch r(t). r(t) = t if -T/2
(a) Show that (10) implies(b) Solve Prob. 8 by applying (a3) to the result of Prob. 7.Data from Prob. 7Represent f(x) as an integral (10).(c) Verify (a2) for f(x) = 1 if 0 < x < Î±
In Table III obtain formula 7 from formula 8. f(w) = F(f) f(x) (1 if -b < x
Using (8), prove that the series has the indicated sum. Compute the first few partial sums to see that the convergence is rapid. Зп 4 cos* x dx 4.
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.f(x) = cos πx (-1/2 < x < 1/2), p = 1
Find the steady-state oscillations of y" + cy' + y = r(t) with c > 0 and r(t) as given. The spring constant is k = 1. Show the details and sketch r(t). (-1 if - TT
These orthogonal polynomials are defined by Heo(1) = 1 andAs is true for many special functions, the literature contains more than one notation, and one sometimes defines as Hermite polynomials the
Find F(f(x)) for f(x) = xe-x if x > 0, f(x) = 0 if x < 0, by (9) and formula 5 in Table III (with Î± = 1). Consider xe-xand e-x. f(w) = F(f) f(x) (1 if -b < x
Represent f(x) as an integral (10). if 0
What function does the series of the cosine terms in Prob. 13 represent The series of the sine terms?Data from Prob. 13Find the Fourier series of as given over one period and sketch and partial sums.
Why does the series in Prob. 11 have no cosine terms?Data from Prob. 11Find the Fourier series of as given over one period and sketch and partial sums. (0 if -2
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.f(x) = 1 - x2/4 (-2 < x < 2), p = 4
Using formulas 2 and 4 in Table II, a value needed for Bessel functions and other applications. Î,(w) = F,(f) f(x) (1 if 0 0) arctan a п! Im = "-a (a > 0) Imaginary part V7 (a +wyn+T Im (a + iw)+1
Find from (4b) and a suitable formula in Table I. fa(w) = F,(f) f(x) < a (1 if 0 0) IT a (a > 0) V2a 2. V T (a2 + w?yw+1 Re = n! (a > 0) -Re (a + iw)+1 Real part Scos x if 0 0) :cos V2a 4a sin ax (a
Find the eigenvalues and eigenfunctions. Verify orthogonality. Start by writing the ODE in the form (1), using Prob. 6. Show details of your work.y" - 2y' + (λ + 1)y = 0, y(0) = 0, y(1) = 0Data from
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details. (x if -1
Find and graph (on common axes) the partial sums up to Sm0 whose graph practically coincides with that of f(x) within graphical accuracy. State m0. On what does the size of m0 seem to
Find the eigenvalues and eigenfunctions. Verify orthogonality. Start by writing the ODE in the form (1), using Prob. 6. Show details of your work.y" + λy = 0, y(0) = y(1), y'(0) = y'(1)Data from
Find and graph (on common axes) the partial sums up to Sm0 whose graph practically coincides with that of f(x) within graphical accuracy. State m0. On what does the size of m0 seem to
Find and graph (on common axes) the partial sums up to Sm0 whose graph practically coincides with that of f(x) within graphical accuracy. State m0. On what does the size of m0 seem to
Represent f(x) as an integral (10). .2 if 0
Find the Fourier transform. Show details. if -1
Does the Fourier cosine transform of f(x) = k = const (0 < x < ∞) exist? The Fourier sine transform?
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work. -1
Find the eigenvalues and eigenfunctions. Verify orthogonality. Start by writing the ODE in the form (1), using Prob. 6. Show details of your work.y" + λy = 0, y(0) = 0, y(L) = 0Data from Prob. 6Show
Find a general solution of the ODE y" + ω2y = r(t) with r(t) as given. Show the details of your work.r(t) = π/4 |cos t| if -π < t < π and r(t + 2π) = r(t), |ω| ≠ 0, 2, 4, · · ·
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details. -|x| f(x) = e-lal (-0
Why are the square errors in Prob. 5 substantially larger than in Prob. 3?Data from Prob. 3Find the trigonometric polynomial F(x) of the form (2) for which the square error with respect to the given
Show that the integral represents the indicated function. Use (5), (10), or (11); the integral tells you which one, and its value tells you what function to consider. Show your work in detail. (7 sin
cos nx, sin nx, cos 2πx/k, sin 2πx/k, cos 2πnx/k, sin 2πnx/k.
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [cosh y, 0, sinh x], S: z = x + y2, 0 ≤ y ≤ x, 0 ≤ x ≤ 1
Why are x̅ and , y̅ in Prob. 25 independent of k?Data from Prob. 25Find the coordinates x̅, y̅ of the center of gravity of a mass of density f(x, y) in the region R. Sketch R, show details.f =
Evaluate them with F or f and C as follows.f = 3x + y + 5z, C: r = [t, cosh t, sinh t], 0 ≤ t ≤ 1. Sketch C.
u = const and v = const on r(u, v) occur if and only if ru • rv = 0. Give examples. Prove it.
Similarity is basic, for instance, in designing numeric methods.(a) By definition, the trace of an n Ã— n matrix A = [Î±jk] is the sum of the diagonal entries,trace A =
Show that
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Findv • ((curl w) × v)
Find the component of a in the direction of b. Make a sketch.a = [8, 2, 0], b = [-4, -1, 0]
Find v such that the resultant of p, q, u, v with p, q, u as in Prob. 24 has no components in x- and y-directions.Data from Prob. 24Find the resultant in terms of components and its magnitude.p =
Find v such that p, q, u in Prob. 21 and v are in equilibrium.Data from Prob. 21Find the resultant in terms of components and its magnitude.p = [2, 3, 0], q = [0, 6, 1], u = [2, 0, -4]
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Deduce the law of cosines by using vectors a, b, and a - b.

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