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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

When is the component of a vector v in the direction of a vector w equal to the component of w in the direction of v?
Solve Prob. 25 if p = [1, 0, 3], Q: (2, 0, 3), and A: (4, 3, 5).Data from Prob. 25Find the moment vector m and m of p = [2, 3, 0] about Q: (2, 1, 0) acting on a line through A: (0, 3, 0). Make a
Given a curve C: r(t), find a tangent vector r'(t), a unit tangent vector u'(t), and the tangent of C at P. Sketch curve and tangent.r(t) = [cos t, sin t, 9t], P: (1, 0, 18π)
Experiments show that in a temperature field, heat flows in the direction of maximum decrease of temperature T. Find this direction in general and at the given point P. Sketch that direction at P as
Given a curve C: r(t), find a tangent vector r'(t), a unit tangent vector u' (t), and the tangent of C at P. Sketch curve and tangent.r(t) = [t, 1/2t2, 1], P: (2, 2, 1)
Find a parametric representation.Straight line through (1, 1, 1) and (4, 0, 2). Sketch it.
Assuming sufficient differentiability, show that(a) curl (u + v) = curl u + curl v(b) div (curl v) = 0(c) curl (fv) = (grad f) × v + f curl v(d) curl (grad f) = 0(e) div (u × v) = v • curl u - u
Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work.v = (x2 + y2 + z2)-3/2 [x, y, z]
A quadratic form Q(x) = xTAx and its (symmetric!) matrix A are called (a) positive definite If Q(x) > 0 for all x ‰ 0, (b) negative definite If Q(x) < 0 for all x ‰ 0, (c)
Let A = [αjk] be an n × n matrix with (not necessarily distinct) eigenvalues λ1, · · · , λn. Show.A Leslie matrix L with positive l12, l13, l21, l32 has a positive eigenvalue.
Find the eigenvalues. Find the corresponding eigenvectors. 3
Are the following sets of vectors linearly independent? Show the details of your work. [4 -1 3], [0 8 1), [1 3 -5], [2 6 1]
If you write the matrix in Example 2 in the form A = [αjk], what is α31? α13? α26? α33?
Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, v2, · · · denote components.)All vectors in Rn with |vj| = 1 for j =
Determine the ranks of the coefficient matrix and the augmented matrix and state how many solutions the linear system will have.In Prob. 24Data from Prob. 24Showing the details, find all solutions or
Verify (3) for the vectors in Probs. 16 and 19.Data from Prob. 16Find the Euclidean norm of the vectors:[1/2 1/3 -1/2 -1/3]TData from Prob. 19Find the Euclidean
Give an application of the matrix in Prob. 2 that makes the form of the inverse obvious.Data from Prob 2Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). sin 20
Showing the details, evaluate: 4 1 0 -2 2. 2.
Showing the details, calculate the following expressions or give reason why they are not defined, whenuTv, uvT 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Prove the formula in Prob. 13.Data from prob 13Verify (AT)-1 = (A-1)T for A in Prob. 1.Data from Prob 1Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). -2.32
Prove the formula in Prob. 11.Data from Prob. 11Verify (A2)-1 = (A-1)2 for A in Prob. 1.Data from Prob. 1Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). -2.32
Showing the details, calculate the following expressions or give reason why they are not defined, whenAT, BT 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.(4 · 3)A, 4(3A), 14B - 3B, 11B Let 4 A 6. B = 3 -3 -2 4 -2
(a) Illustrate (10d) by simple examples.(b) Prove (10d).
If a 12 × 12 matrix A shows the distances between 12 cities in kilometers, how can you obtain from A the matrix B showing these distances in miles?
Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). 0.1 -0.4 2.5
What is the main diagonal of A in Example 1? Of A and B in Example 3?
How many different entries can a 4 × 4 skew-symmetric matrix have? An n × n skew-symmetric matrix?
Solve the linear system given explicitly or by its augmented matrix. Show details. [3.0 -0.5 0.6 1.5 4.5 6.0
Showing the details of your work, find L(f) if f(t) equals:te-t cos t
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.xy" + y' + 36y = 0 (12√x = z)
Find a general solution by conversion to a single ODE.The system in Prob. 8.Data from Prob. 8y'1 = 8y1 - y2y'2 = y1 + 10y2
Find the inverse transform. Show the details of your work. a(s + k) + bT (s + k)? + 7?
Find the inverse transform. Show the details of your work. do (s + 1)2 (s + 1)3
Find the inverse transform. Show the details of your work. s2 - 2s – 3 4.
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work. 4s + 32 - 16 s2 - 16
Using Theorem 3, find f(t) if L(F) equals: ,2
Model and solve by the Laplace transform:Show that, by Kirchhoff€™s Voltage Law, the currents in the network in Fig. 153 are obtained from the systemSolve this system, assuming that R = 10
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = 4y2 - 8 cos 4t, y'2 = -3y1 - 9 sin 4t, y1(0) = 0, y2(0) = 3
Find and graph or sketch the solution of the IVP. Show the details.y" + 16y = 4δ(t - 3π), y(0) = 2, y’(0) = 0
Find:(cos ωt) * (cos ωt)
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.cos 4t (0 < t
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion. Show all details.y" + 9y = 10e-t, y(0) = 0, y'(0) = 0
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 + y2 = 0, y1 + y'2 = 2 cos t, y1(0) = 1, y2(0) = 0
(a) Text, take a rectangular wave of area 1 from 1 to 1 + k. Graph the responses for a sequence of values of k approaching zero, illustrating that for smaller and smaller k those curves approach the
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.t (0 < t <
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.(α - bt)2
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y' + 2y = 0, y(0) = 1.5
Show that Iv(x) is real for all real x (and real v), Iv ≠ 0 for all real x ≠ 0, and I-n(x) = In(x), where n is any integer.
Apply the power series method. Do this by hand, not by a CAS, to get a feel for the method, e.g., why a series may terminate, or has even powers only, etc. Show the details.xy' - 3y = k (= const)
Apply the power series method. Do this by hand, not by a CAS, to get a feel for the method, e.g., why a series may terminate, or has even powers only, etc. Show the details.(1 + x)y' = y
If a system has a center as its critical point, what happens if you replace the matrix A by A∼ = A + kI with any real number k ≠ 0 (representing measurement errors in the diagonal entries)?
Find a general solution. Determine the kind and stability of the critical point.y'1 = 4y2y'2 = -4y1
What happens to the critical point in Example 1 if you introduce τ = -t as a new independent variable?
Find the location and type of all critical points by first converting the ODE to a system and then linearizing it.y" + 9y + y2 = 0
A bar with heat generation of constant rate H( >0) is modeled by ut = c2uxx + H. Solve this problem if L = π and the ends of the bar are kept at 0°C. Set u = v - Hx(x - π)/(2c2).
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,
Show that among all rectangular membranes of the same area A = αb and the same c the square membrane is that for which u11 has the lowest frequency.
Find the potential in the interior of the sphere r = R = 1 if the interior is free of charges and the potential on the sphere isf (Φ) = 1 - cos2 Φ
Find the type, transform to normal form, and solve. Show your work in detail.uxx - 6uxy + 9uyy = 0
Nonzero initial velocity is more of theoretical interest because it is difficult to obtain experimentally. Show that for (17) to satisfy (9b) we must havewhere Km =
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,
Transform to normal form and solve:uxx + 6uxy + 9uyy = 0
Show that the Tricomi equation yuxx + uyy = 0 is of mixed type. Obtain the Airy equation G" - yG = 0 from the Tricomi equation by separation.
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
Find the steady-state solutions (temperatures) in the square plate in Fig. 297 with Î± = 2 satisfying the following boundary conditions. Graph isotherms.(a) u = 80 sin Ï€x on
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 the potentials exterior to the sphere in Probs. 16 and 19.Data from Prob. 16Find the potential in the interior of the sphere r = R = 1 if the interior is free of charges and the potential on the
Find the steady-state temperature in the plate in Prob. 21 if the lower side is kept at U0°C the upper side at U1°C and the other sides are kept at 0°C. Split into two problems in which
(a) Show that ez is entire. What about e1/z? ez̅? ex(cos ky + i sin ky)? (Use the Cauchy–Riemann equations.)(b) Find all z such that(i) ez is real.(ii) |e-z| < 1.(iii) ez̅ = e̅z̅.(c) It
Find the value of the derivative of(1.5z + 2i)/(3iz - 4) at any z. Explain the result.
Find all solutions and graph some of them in the complex plane.ez = 4 + 3i
Find the radius of convergence in two ways:(a) Directly by the Cauchy€“Hadamard formula in Sec. 15.2.(b) From a series of simpler terms by using Theorem 3 or Theorem 4.
Find the radius of convergence in two ways:(a) Directly by the Cauchy€“Hadamard formula in Sec. 15.2.(b) From a series of simpler terms by using Theorem 3 or Theorem 4.
Find the radius of convergence in two ways:(a) Directly by the Cauchy€“Hadamard formula in Sec. 15.2.(b) From a series of simpler terms by using Theorem 3 or Theorem 4. 00
Prove that the series converges uniformly in the indicated region.
Find the center and the radius of convergence.
Find all the singularities in the finite plane and the corresponding residues. Show the details.e1/(1-z)
Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons.tan πz
Sketch or graph the given region and its image under the given mapping.|z| < 1/2, Im z > 0, w = 1/z
Find the fixed points. aiz – 1 a + 1 z+ ai
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. sin z Zo Zo = 17 (z – m)3*
Evaluate (counterclockwise). Show the details. el/ dz, C: the unit circle
Evaluate the following integrals and show details of your work. cos 2x dx (x2 + 1)2
Integrate counterclockwise around C. Show the details.15z + 9/z3 - 9z, C:|z| = 4
Find the LFT that maps the given three points onto the three given points in the respective order.-3/2, 0, 1 onto 0, 3/2, 1
Find an LFT that maps |z| ≤ 1 onto |w| ≤ 1 so that z = i/2 is mapped onto w = 0. Sketch the images of the lines x = const and y = const.
Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons.z3 exp(1/z- 1)
Sketch or graph the given region and its image under the given mapping.-1 ≤ x ≤ 2, -π < y < π, w = ez
Find the images of the lines y = k = const under the mapping w = cos z.
Evaluate (counterclockwise). Show the details. dz, C: |z – 1| = 2 Jczt – 223
Evaluate the following integrals and show details of your work. 00 cos 4x – dx x* + 5x2 + 4
Integrate counterclockwise around C. Show the details.cot 4z, C:|z| = 3/4
Find an analytic function that maps the second quadrant of the z-plane onto the interior of the unit circle in the w-plane.
Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons.1/(cos z - sin z)
Sketch or graph the given region and its image under the given mapping.1/2 ≤ |z| ≤ 1, 0 ≤ θ < π/2, w = Ln z
Find and sketch or graph the image of the given region under the mapping w = cos z.0 < x < 2π, 1/2 < y < 1
Find all LFTs with fixed point(s).Without any fixed points
Evaluate the following integrals and show details of your work. dx .3 8 -x*
Evaluate by the methods of this chapter. Show details. .27 sin 0 3 + cos 0

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