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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find all the singularities in the finite plane and the corresponding residues. Show the details.z4/z2 - iz + 2
Evaluate the following integrals and show details of your work. 00 dx (1 + x2,3 -0
Find the LFT that maps the given three points onto the three given points in the respective order.0, -i, i onto -1, 0, ∞
(a) Show that if f(z) has a zero of order n > 1 at z = z0, then f'(z) has a zero of order n - 1 at z0.(b) Poles and zeros. Prove Theorem 4.(c) Show that the points at which a nonconstant analytic
Sketch or graph the given region and its image under the given mapping.1 < |z| < 3, 0 < Arg z < π/2, w = z3
Find and sketch or graph the image of the given region under w = sin z.-π/4 < x < π/4, 0 < y < 1
Find the fixed points.w = z - 3i
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. 1 Zo ?(z – i)'
Evaluate the following integrals and show details of your work. 00 dx (x2 – 2x + 5)2 00
Integrate counterclockwise around C. Show the details.e2/z, C:|z - 1 - i| = 2
Find the LFT that maps the given three points onto the three given points in the respective order.0, 2i, -2i onto -1, 0, ∞
Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons. e-i + z -i (z - i)3
Sketch or graph the given region and its image under the given mapping.x ≥ 1, w = 1/z
Find and sketch or graph the image of the given region under w = sin z.0 < x < π/6, -∞ < y < ∞
Find the fixed points.w = αz + b
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. az Zo = b – b' z- b'
Evaluate (counterclockwise). Show the details. z - 23 - dz, C: |z – 2 – i 22 - 4z – 5 = 3.2
Evaluate the following integrals and show details of your work. 00 .2 + 1 dx x* + 1
Integrate counterclockwise around C. Show the details.5z3/z2 + 4, C:|z - i| = πi/2
Find the LFT that maps the given three points onto the three given points in the respective order.-1, 0, 1 onto 1, 1 + i, 1 + 2i
Find the Maclaurin series and its radius of convergence. exp (z) exp (-1²) dt t
If z(x, y) = 300 - x2 - 9y2 [meters] gives the elevation of a mountain at sea level, what is the direction of steepest ascent at P: (4, 1)?
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, t2, t3], P: (1, 1, 1)
Find the area of the quadrangle Q whose vertices are the midpoints of the sides of the quadrangle P with vertices A: (2, 1, 0), B: (5, -1, 0), C: (8, 2, 0), and D: (4, 3, 0). Verify that Q is a
When is the moment of a force equal to zero?
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Find the angles of the triangle with vertices A: (0, 0, 2), B: (3, 0, 2), and C: (1, 1, 1). Sketch the triangle.
Find the unit vector in the direction of the resultant in Prob. 24.Data from Prob. 24Find the resultant in terms of components and its magnitude.p = [-1, 2, -3], q = [1, 1, 1], u = [1, -2, 2]
Find the length and sketch the curve.r(t) = [4 cos t, 4 sin t, 5t] from (4, 0, 0) to (4, 0, 10π).
Find the plane through the points A: (1, 2, 1/4), B: (4, 2, -2), and C: (0, 8, 4).
Find the velocity, speed, and acceleration of the motion given by r(t) = [3 cos t, 3 sin t, 4t] (t = time) at the point P: (3/√2, 3/√2, π).
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Find the distance of the point A: (1, 0, 2) from the plane P: 3x + y + z = 9. Make a sketch.
Find the length and sketch the curve.r(t) = [α cos3 t, α sin3 t], total length.
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Find grad f and f grad f at P: (2, 7, 0)
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.For what c are 3x + z =
If |p| = 6 and |q| = 4, what can you say about the magnitude and direction of the resultant? Can you think of an application to robotics?
(a) Evaluate ∫C f(z) dz by Theorem 1 and check the result by Theorem 2, where:(i) f(z) = z4 and C is the semicircle |z| = 2 from 2i to 2i in the right half-plane,(ii) f(z) = e2z and C is the
Find the type, transform to normal form, and solve. Show your work in detail.uxx + 2uxy + 10uyy = 0
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) = cos Φ
Solve for u = u (x, y):uxx + 25u = 0
Find the temperature in Prob. 11 with L = Ï€, c = 1, andf (x) = cos 2xData from Prob. 11€œAdiabatic€ means no heat exchange with the neighborhood, because the bar is
(a) Verify that u(x, t) = v(x + ct) + w(x - ct) with any twice differentiable functions v and w satisfies (1).(b) Verify that u = 1/√x2 + y2 + z2 satisfies (6) and u = ln (x2 + y2) satisfies (3).
A small drum should have a higher fundamental frequency than a large one, tension and density being the same. How does this follow from our formulas?
(a) Write a program for calculating the Am’s in Example 1 and extend the table to m = 15. Verify numerically that αm ≈ (m - 1/4)π and compute the error for m = 1, · · ·, 10.(b) Graph the
Find the deflection u(x, y, t) of the square membrane of side π and c2 = 1 for initial velocity 0 and initial deflection0.01 sin x sin y
Find the electrostatic potential between coaxial cylinders of radii r1 = 2 cm and r2 = 4 cm kept at the potentials U1 = 220 V and U2 = 140 V, respectively.
Find the temperature w(x, t) in a semi-infinite laterally insulated bar extending from x = 0 along the x-axis to infinity, assuming that the initial temperature is 0, w(x, t) †’ 0 as x
Find the type, transform to normal form, and solve. Show your work in detail.uxx - 16uyy = 0
Show that the only solution of Laplace’s equation depending only on r = √x2 + y2 is u = c ln r + k.
Find the temperature w(x, t) in a semi-infinite laterally insulated bar extending from x = 0 along the x-axis to infinity, assuming that the initial temperature is 0, w(x, t) †’ 0 as x
Write a program that gives you four numerically equal λmn in Example 1, so that four different Fmn correspond to it. Sketch the nodal lines of F18, F81, F47, F74 in Example 1 and similarly for
The electrostatic potential satisfies Laplace’s equation ∇2u = 0 in any region free of charges. Also the heat equation ut = c2 ∇2u (Sec. 12.5) reduces to Laplace’s equation if the
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Heat Equation (2) with suitable cu = e-9t sin ωx
If the ends x = 0 and x = L of the bar in the text are kept at constant temperatures U1 and U2 respectively, what is the temperature u1(x) in the bar after a long time (theoretically, as t → ∞)?
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), whereVerify that u in the solution of Prob. 7 satisfies the initial condition.Data from Prob.
Solve by Laplace Transforms + 25w, + 100 w (x, 0) = 0 if x 0, w;(x, 0) = 0if t2 0, w (0, t) = sin t ift20 100 at хр
Represent f (x, y) by a series (15), wheref (x, y) = xy (α - x) (b - y), α and b arbitrary
The electrostatic potential satisfies Laplace’s equation ∇2u = 0 in any region free of charges. Also the heat equation ut = c2 ∇2u reduces to Laplace’s equation if the temperature u is
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Heat Equation (2) with suitable cu = e-t sin x
Solve by Laplace Transforms w(x, 0) = 1, w(0, t) = 1 + 2x- дw aw at = 2x, ах
Represent f (x, y) by a series (15), wheref (x, y) = x, α = b = 1
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Wave Equation (1) with suitable cu = sin kct cos kx
(a) Show that un= rncos nÎ¸, un= rnsin nÎ¸, n = 0, 1,· · ·, are solutions of Laplace€™s equation ˆ‡2u = 0 with ˆ‡2u given by
Find the surfaces on which u1, u2, u3 in (16) are zero.
Solve by Laplace Transforms дw дw 3D 1, w(0, t) %3D 1 х, w(x, 0) at дх
Represent f (x, y) by a series (15), wheref (x, y) = 1, α = b = 1
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Wave Equation (1) with suitable cu = x2 + t2
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref(x) = 1 if |x| < α and 0 otherwise
How would the motion of the string change if Assumption 3 were violated? Assumption 2? The second part of Assumption 1? The first part? Do we really need all these assumptions?
Which part of Assumption 2 cannot be satisfied exactly? Why did we also assume that the angles of inclination are small?
Find (a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. L L
Find (a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. kIN
Write the Fourier matrix F for a sample of eight values explicitly.
How does the minimum square error change if you multiply f(x) by a constant k?
Find(a) The Fourier cosine series(b) The Fourier sine series. Sketch f(x) and its two periodic extensions. Show the details. 4 2.
Obtain the Fourier series in Prob. 8 from that in Prob. 17.Data from Prob. 8Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.Data from
Using Prob. 11, show that 1 + 1/4 + 1/9 + 1/16 + · · · = 1/6 π2.Data from Prob. 11Is the given function even or odd or neither even nor odd? Find its Fourier series. Show details of your work.f
Graph and discuss outputs of y" + cy' + ky = r(t) with r(t) as in Example 1 for various c and k with emphasis on the maximum Cn and its ratio to the second largest |Cn|.
Show that solutions of (22) satisfying (27) are (see Fig. 310)
Consider a long cable or telephone wire (Fig. 315) that is imperfectly insulated, so that leaks occur along the entire length of the cable. The source S of the current i(x, t) in the cable is at x =
Find steady-state temperatures in the rectangle in Fig. 296 with the upper and left sides perfectly insulated and the right side radiating into a medium at 0°C according to ux(Î±, y)
Show that the solutions z = z(x, y) of yzx = xzy represent surfaces of revolution. Give examples. Use polar coordinates r, θ and show that the equation becomes zθ = 0.
Show that the following membranes of area 1 with c2 = 1 have the frequencies of the fundamental mode as given (4-decimal values). Compare.Semicircle: 3.832/√8π = 0.7643
Show that i2 = -1, i3 = -i, i4 = 1, i5 = i, · · · and 1/i = -i, 1/i2 = -1, 1/i3 = i, · · · .
Find the potential in the following charge-free regions.In the interior of a sphere of radius 1 kept at the potential f(Φ) = cos 3Φ + 3 cos Φ (referred to our usual spherical coordinates).
Determine and sketch or graph the sets in the complex plane given by0 < |z| < 1
Find ez in the form u + iv and |ez| if z equals3 + 4i
Are the following functions analytic? Use (1) or (7).f(z) = izz̅
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.-4 + 4i У pl+i л/4
What happens to a quotient if you take the complex conjugates of the two numbers? If you take the absolute values of the numbers?
Formulas for hyperbolic functionsShow thatcosh (z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2sinh (z1 + z2) = sinh z1 cosh z2 + cosh z1 sinh z2
Multiplication by i is geometrically a counterclockwise rotation through π/2 (90°). Verify this by graphing z and iz and the angle of rotation for z = 1 + i, z = -1 + 2i, z = 4 - 3i.
Are the following functions analytic? Use (1) or (7).f(z) = ex (cos y - i sin y)
Find ez in the form u + iv and |ez| if z equals0.6 - 1.8i
Determine and sketch or graph the sets in the complex plane given by-π < Im z < π
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.-5 У pl+i л/4
Verify (9) for z1 = -11 + 10i, z2 = -1 + 4i.
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. V3 - 10i -V3 + 5i У pl+i л/4
Determine and sketch or graph the sets in the complex plane given byRe (1/z) < 1
Find Ln z when z equals4 + 4i
Find ez in the form u + iv and |ez| if z equals11πi/2

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