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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [0,
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).4y dx + z dy + (y - 2z) dz
Make Möbius strips from long slim rectangles R of grid paper (graph paper) by pasting the short sides together after giving the paper a half- twist. In each case count the number of parts obtained
Evaluate them with F or f and C as follows.F = [x + y, y + z, z + x], C: r = [4 cos t, sin t, 0], 0 ≤ t ≤ π
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Find Ix, Iy, I0 of a mass of density f(x,y) = 1 in the region R in the figures, which the engineer is likely to need, along with other profiles listed in engineering handbooks. R as in Prob. 13. Data
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Sphere x2 + (y + 2.8)2 + (z - 3.2)2 = 2.25
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).(cos (x2 + 2y2 + z2)) (2x dx + 4y dy + 2z dz)
Justify the following formulas for the mass Mand the center of gravity (x̅, y̅, z̅) of a lamina S of density (mass per unit area) σ(x, y, z) in space:
Evaluate them with F or f and C as follows.f = xyz, C: r = [4t, 3t2, 12t], -2 ≤ t ≤ 2. Sketch C.
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Hyperbolic cylinder x2 - y2 = 1
Find Ix, Iy, I0 of a mass of density f(x,y) = 1 in the region R in the figures, which the engineer is likely to need, along with other profiles listed in engineering handbooks. I EIN NIG NIG H NIR
Given a mass of density 1 in a region T of space, find the moment of intertia about the x-axisIx = ∫∫T∫ (y2 + z2) dx dy dz.The box -α ≤ x ≤ α, -b ≤ y ≤ b, -c ≤ z ≤ c
Justify the following formulas for the moments of inertia of the lamina in Prob. 19 about the x-, y-, and z-axes, respectively: Data from Prob. 19 Justify the following formulas for the mass Mand
Find a formula for the moment of inertia of the lamina in Prob. 20 about the line y = x, z = 0. Data form Prob. 20 Justify the following formulas for the moments of inertia of the lamina in Prob. 19
Find 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 = xy, R the triangle with vertices (0, 0), (2, 0), (2, 2)
Given a mass of density 1 in a region T of space, find the moment of intertia about the x-axisIx = ∫∫T∫ (y2 + z2) dx dy dz.The cylinder y2 + z2 ≤ α2, 0 ≤ x ≤ h
Find 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 = x2 + y2, R: x2 + y2 ≤ α2, y ≥ 0
Find the moment of inertia of a lamina S of density 1 about an axis B, whereS: x2 + y2 = z2, 0 ≤ z ≤ h, B: the z-axis
Find 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 = x2, R: -1 ≤ x ≤ 2, x2 ≤ y ≤ x + 2. Why is x̅ > 0?
Given a mass of density 1 in a region T of space, find the moment of intertia about the x-axisIx = ∫∫T∫ (y2 + z2) dx dy dz.The cone y2 + z2 ≤ x2, 0 ≤ x ≤ h
Using Steiner’s theorem, find the moment of inertia of a mass of density 1 on the sphere S: x2 + y2 + z2 = 1 about the line K: x =1, y = 0 from the moment of inertia of the mass about a suitable
Find 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 = ky, k > 0, arbitrary, 0 ≤ y ≤ 1 - x2, 0 ≤ x ≤ 1
Show that for a solid of revolution, Ix = π/2 ∫h0 r4 (x) dx. Solve Probs. 20–23 by this formula.
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [αx, by, cz], S the sphere x2 + y2 + z2 = 36
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [y + z, 20y, 2z3], S the surface of 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, 0 ≤ z ≤ y
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [ex, ey, ez], S the surface of the box |x| ≤ 1, |y| ≤ 1, |z| ≤ 1
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [y2, x2,z2], S: r = [u, u2, v], 0 ≤ u ≤ 2, -2 ≤ v ≤ 2
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [x + z, y + z, x + y], S the sphere of radius 3 with center 0
Derive the formula for Cn from An and Bn.
Are the following functions even or odd or neither even nor odd?ex, e-|x|, x3 cos nx, x2 tan πx, sinh x - cosh x
Find the cosine transform f̂c of f(x) = 1 if 0 < x < 1, f (x) = -1 if 1 < x < 2, f (x) = 0 if x > 2.
What is a Fourier series? A Fourier cosine series? A half-range expansion? Answer from memory.
The fundamental period is the smallest positive period. Find it forcos x, sin x, cos 2x, sin 2x, cos πx, sin πx, cos 2πx, sin 2πx
Show that 1/i = -i, e-ix = cos x - i sin x, eix + e-ix = 2 cos x, eix - e-ix = 2i sin x, eikx = cos kx + i sin kx.
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. cos
Showing the details, develop63x5 - 90x3 + 35x
What are the Euler formulas? By what very important idea did we obtain them?
Explain the role of the Bn’s. What happens if we let c → 0?
Are the following functions even or odd or neither even nor odd?Sums and products of even functions
Find f̂c(w) for f(x) = x if 0 < x < 2, f(x) = 0 if x > 2.
How did we proceed from 2π-periodic to general periodic functions?
Find the trigonometric polynomial F(x) of the form (2) for which the square error with respect to the given f(x) on the interval -π < x < π is minimum Compute the minimum value for N = 1, 2,
If f(x) and g(x) have period p, show that h(x) = αf(x) + bg(x) (α, b, constant) has the period p. Thus all functions of period p form a vector space.
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details.
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.
Showing the details, develop1 - x4
Show that if the functions y0(x), y1(x), · · · form an orthogonal set on an interval α ≤ x ≤ b (with r(x) = 1), then the functions y0(ct + k), y1(ct + k), · · ·, c > 0, form an
Can a discontinuous function have a Fourier series? A Taylor series? Why are such functions of interest to the engineer?
Find the trigonometric polynomial F(x) of the form (2) for which the square error with respect to the given f(x) on the interval -π < x < π is minimum Compute the minimum value for N = 1, 2,
Some of the An in Example 1 are positive, some negative. All Bn are positive. Is this physically understandable?
Show that the functions Pn(cos θ), n = 0, 1, · · ·, from an orthogonal set on the interval 0 ≤ θ ≤ π with respect to the weight function sin θ.
Are the following functions even or odd or neither even nor odd?Absolute values of odd functions
What do you know about convergence of a Fourier series? About the Gibbs phenomenon?
Find f̂c(w) for f(x) = x2 if 0 < x < 1, f(x) = 0 if x > 1.
Find the trigonometric polynomial F(x) of the form (2) for which the square error with respect to the given f(x) on the interval -π
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details.
Show that f = const is periodic with any period but has no fundamental period.
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.
Showing the details, developProve that if f(x) is even (is odd, respectively), its Fourier–Legendre series contains only Pm(x) with even m (only Pm(x) with odd m, respectively). Give examples.
Find a general solution of the ODE y" + ω2y = r(t) with r(t) as given. Show the details of your work.r (t) = sin αt + sin βt, ω2 ≠ α2, β2
The output of an ODE can oscillate several times as fast as the input. How come?
Sketch or graph f(x) which for -π < x < π is given as follows.f(x) = |x|
Find a general solution of the ODE y" + ω2y = r(t) with r(t) as given. Show the details of your work.r (t) = sin t, ω = 0.5, 0.9, 1.1, 1.5, 10
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(10) = 0Data from Prob.
Are the following functions even or odd or neither even nor odd?Find all functions that are both even and odd.
What is approximation by trigonometric polynomials? What is the minimum square error?
Does the Fourier cosine transform of x-1 sin x (0 < x < ∞) exist? Of x-1 cos x? Give reasons.
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details.
Sketch or graph f(x) which for -π < x < π is given as follows.f(x) = |sin x|, f (x) = sin |x|
Represent f(x) as an integral (10).
Showing the details, developWhat happens to the Fourier–Legendre series of a polynomial f(x) if you change a coefficient of f(x)? Experiment. Try to prove your answer.
What is a Fourier integral? A Fourier sine integral? Give simple examples.
Sketch or graph f(x) which for -π < x < π is given as follows.f(x) = e-|x|, f(x) = |e-x|
Find a general solution of the ODE y" + ω2y = r(t) with r(t) as given. Show the details of your work.What kind of solution is excluded in Prob. 8 by |ω| ≠ 0, 2, 4, · · ·?Data from Prob. 8r (t)
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.
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.
Find Fs(e-αx), α > 0, by integration.
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
Show that the minimum square error (6) is a monotone decreasing function of N. How can you use this in practice?
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details.
Sketch or graph f(x) which for -π
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 |sin t| if 0 < t < 2π and r(t + 2π) = r(t), |ω| ≠ 0, 2, 4, · · ·
What are Sturm–Liouville problems? By what idea are they related to Fourier series?
Sketch or graph f(x) which for -π
Find a general solution of the ODE y" + ω2y = r(t) with r(t) as given. Show the details of your work.
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'/x)' + (λ + 1)y/x3 = 0, y(1) = 0, y(eπ) = 0.
Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.f (x) = x2 (-1 < x < 1), p = 2
Find fs (w) for f(x) = x2 if 0 < x < 1, f(x) = 0 if x > 1.
Find the Fourier series of as given over one period and sketch and partial sums.
Using (8), prove that the series has the indicated sum. Compute the first few partial sums to see that the convergence is rapid.
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details.
Review integration techniques for integrals as they are likely to arise from the Euler formulas, for instance, definite integrals of x cos nx, x2 sin nx, e-2x cos nx, etc.
Write a program for solving the ODE just considered and for jointly graphing input and output of an initial value problem involving that ODE. Apply the program to Probs. 7 and 11 with initial values
Using (8), prove that the series has the indicated sum. Compute the first few partial sums to see that the convergence is rapid.
Find the Fourier series of the given function f(x), which is assumed to have the period 2π. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin
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 steady-state oscillations of y" + cy' + y = r(t) with c > 0 and r(t) as given. Note that the spring constant is k = 1. Show the details. In Probs. 14–16 sketch r(t).

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