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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Are the following functions analytic? Use (1) or (7).f(z) = cos x cosh y - i sin x sinh y
How are general powers defined? Give an example. Convert it to the form x + iy.
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:Re (z21), (Re z1)2
Find, in the form u + iv,cosh (-1 + 2i), cos (-2 - i)
Determine the principal value of the argument and graph it as in Fig. 325. -1 + i
Write a report by formulating the corresponding portions of the text in your own words and illustrating them with examples of your own.
ln z is more complicated than ln x. Explain. Give examples.
Find Ln z when z equals0.6 + 0.8i
Are the following functions analytic? Use (1) or (7).f(z) = 3π2/(z3 + 4π2z)
Discuss how ez, cos z, sin z, cosh z, sinh z, are related.
Derive the following laws for complex numbers from the corresponding laws for real numbers.z1 + z2 = z2 + z1, z1z2 = z2z1 (Commutative laws)(z1 + z2) + z3 = z1 + (z2 + z3), (Associative
Find, in the form u + iv,cos i, sin i
State the Cauchy–Riemann equations. Why are they of basic importance?
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. 1 + 1/2πi
Determine and sketch or graph the sets in the complex plane given byRe z ≥ -1
Find Ln z when z equals4 - 4i
Can a function be differentiable at a point without being analytic there? If yes, give an example.
Show that z = x + iy is pure imaginary if and only if z̅ = -z.
Harmonic functions verify by differentiation that Im cos z and Re sin z are harmonic.
What is an analytic function of a complex variable?
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.
Determine and sketch or graph the sets in the complex plane given by|arg z| < 1/4π
Find Ln z when z equals-11
Are the following functions analytic? Use (1) or (7).f(z) = Re (z2) - i Im (z2)
Find ez in the form u + iv and |ez| if z equals2 + 3πi
Formulas for hyperbolic functionsShow thatProve that cos z, sin z, cosh z, and sinh z are entire.
State the definition of the derivative from memory. Explain the big difference from that in calculus.
Formulas for hyperbolic functionsShow thatcosh2 z - sinh2 z = 1, cosh2 z + sinh2 z = cosh 2z
Write the two numbers in Prob. 1 in polar form. Find the principal values of their arguments.Data from Prob. 1Divide 15 + 23i by -3 + 7i. Check the result by multiplication.
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.2i, -2i y 1 1/4 pl+i 1 8
Determine and sketch or graph the sets in the complex plane given byπ < |z - 4 + 2i| < 3π
Find ez in the form u + iv and |ez| if z equals2πi(1 + i)
Are the following functions analytic? Use (1) or (7).f(z) = e-2x(cos 2y - i sin 2y)
Divide 15 + 23i by -3 + 7i. Check the result by multiplication.
Find the potential in the following charge-free regions.Between two coaxial circular cylinders of radii r0 and r1 kept at the potentials u0 and u1, respectively. Compare with Prob. 38.Data from Prob.
Show that the following membranes of area 1 with c2 = 1 have the frequencies of the fundamental mode as given (4-decimal values). Compare.Quadrant of circle: α21/(4√π) = 0.7244(α21 = 5.13562 =
Show that the following membranes of area 1 with c2 = 1 have the frequencies of the fundamental mode as given (4-decimal values). Compare.Rectangle with sides 1:2:√5/8 = 0.7906
Show that the following membranes of area 1 with c2 = 1 have the frequencies of the fundamental mode as given (4-decimal values). Compare.Square: 1/√2 = 0.7071
Show that u11 represents the fundamental mode of a semicircular membrane and find the corresponding frequency when c2 = 1 and R = 1.
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/ρA (E =
These vibrations in the direction of the x-axis are modeled by the wave equation utt = c2uxx, c2 = E/ρ. If the rod is fastened at one end, x = 0, and free at the other, x = L, we have u(0, t) = 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.Circle: α1/(2√π) = 0.6784
Let f(x, y) = u(x, y, 0) be the initial temperature in a thin square plate of side π with edges kept at 0°C and faces perfectly insulated. Separating variables, obtain from ut = c2∇2u the
Let f(x, y) = u(x, y, 0) be the initial temperature in a thin square plate of side π with edges kept at 0°C and faces perfectly insulated. Separating variables, obtain from ut = c2∇2u the
Find the temperature distribution in a laterally insulated bar of length π with c2 = 1 for the adiabatic boundary condition and initial temperature:2π - 4|x - 1/2π|
Find the temperature distribution in a laterally insulated bar of length π with c2 = 1 for the adiabatic boundary condition and initial temperature:100 cos 2x
Find the temperature distribution in a laterally insulated bar of length π with c2 = 1 for the adiabatic boundary condition and initial temperature:3x2
Find the temperature distribution in a laterally insulated thin copper bar (c2 = K/(σρ) = 1.158 cm2/sec) of length 100 cm and constant cross section with endpoints at x = 0 and 100 kept at 0°C and
Find the temperature distribution in a laterally insulated thin copper bar (c2 = K/(σρ) = 1.158 cm2/sec) of length 100 cm and constant cross section with endpoints at x = 0 and 100 kept at 0°C and
Find the temperature distribution in a laterally insulated thin copper bar (c2 = K/(σρ) = 1.158 cm2/sec) of length 100 cm and constant cross section with endpoints at x = 0 and 100 kept at 0°C and
Let r, θ, Φ be spherical coordinates. If u(r, θ, Φ) satisfies ∇2u = 0, show that v(r, θ, Φ) = u(1/r, θ, Φ)/r satisfies ∇2v = 0.
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 steady state temperature in the plate in Prob. 21 with the upper and lower sides perfectly insulated, the left side kept at 0°C, and the right side kept at f(y)°C. Data from Prob. 21 The
Sketch the intersections of the equipotential surfaces in Prob. 16 with xz-plane.Data from Prob. 16Find the potential in the interior of the sphere r = R = 1 if the interior is free of charges and
Find and sketch or graph (as in Fig. 288 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length π, extending from x = 0 to x = π, and c2 = T/ρ = 4 starting with velocity zero and
Show that the boundary condition leads to k = kmn = αmn/R where s = αmn is the mth positive zero of Jn(s).
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,
The faces of the thin square plate in Fig. 297 with side α = 24 are perfectly insulated. The upper side is kept at 25°C and the other sides are kept at 0°C. Find the steady-state temperature u(x,
Show that in Prob. 17 the potential exterior to the sphere is the same as that of a point charge at the origin.Data from Prob. 17Find the potential in the interior of the sphere r = R = 1 if the
Transform to normal form and solve:uxx - 4uyy = 0
Show that Q(θ) must be periodic with period 2π and, therefore, n = 0, 1, 2, · · · in (25) and (26). Show that this yields the solutions Qn = cos nθ, Q*n = sin nθ, Wn = Jn(kr), n = 0, 1,
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) = 10 cos3 Φ - 3 cos2 Φ - 5 cos Φ - 1
Show that forced vibrations of a membrane are modeled by the PDE utt = c2∇2u + P/ρ, where P(x, y, t) is the external force per unit area acting perpendicular to the xy-plane.
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 substitution of u = F(r, θ)G(t) into the wave equation (6), that is,gives an ODE and a PDEShow that the PDE can now be separated by substituting F = W(r)Q(θ), giving (22) Utt = 2 ( 14 + =
Find the potential in the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 if the upper side is kept at the potential 1000 sin 1/2πx and the other sides are grounded.
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 2Φ
Find the deflection of the membrane of sides α and b with c2 = 1 for the initial deflection
Find the potential in the rectangle 0 ≤ x ≤ 20, 0 ≤ y ≤ 40 whose upper side is kept at potential 110 V and whose other sides are grounded.
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,
Is it possible that for fixed c and R two or more um with different nodal lines correspond to the same eigenvalue? (Give a reason.)
The heat flux of a solution u(x, t) across x = 0 is defined by Φ(t) = -Kux(0, t). Find Φ(t) for the solution (9). Explain the name. Is it physically understandable that Φ goes to 0 as t → ∞?
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/ρA (E =
Find the type, transform to normal form, and solve. Show your work in detail.uxx - 4uxy + 5uyy = 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) = 1
Find eigenvalues of the rectangular membrane of sides α = 2 and b = 1 to which there correspond two or more different (independent) eigenfunctions.
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/ρA (E =
Find the temperature in Prob. 11 with L = π, c = 1, and f (x) = x Data from Prob. 11 “Adiabatic” means no heat exchange with the neighborhood, because the bar is completely insulated, also at
Following function is important in applied mathematics and physics (probability theory and statistics, thermodynamics, etc.) and fits our present discussion. Regarding it as a typical case of a
Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph u(x, t) as in Fig. 291 in
Verify that the function u (x, y) = α ln (x2 + y2) + b satisfies Laplace’s equation (3) and determine α and b so that u satisfies the boundary conditions u = 110 on the circle x2 + y2 = 1 and u =
Find a formula for the tension required to produce a desired fundamental frequency f1 of a drum.
Find the temperature in Prob. 11 with L = π, c = 1, and f (x) = 1 - x/π Data from Prob. 11 “Adiabatic” means no heat exchange with the neighborhood, because the bar is completely insulated,
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/ρA (E =
Find the type, transform to normal form, and solve. Show your work in detail.xuxx - yuxy = 0
What are the analogs of Probs. 12 and 13 in heat conduction?Data from Prob. 12Find the electrostatic potential between coaxial cylinders of radii r1 = 2 cm and r2 = 4 cm kept at the potentials U1 =
Explain how the Laplace transform applies to PDEs.
Do Prob. 3 for the membrane with α = 4 and b = 2.Data from Prob. 3Determine and sketch the nodal lines of the square membrane for m = 1, 2, 3, 4 and n = 1, 2, 3, 4.
If the surface of the ball r2 = x2 + y2 + z2 ≤ R2 is kept at temperature zero and the initial temperature in the ball is f(r), show that the temperature u(r, t) in the ball is a solution of ut =
Find the deflection u(x, t) of the string of length L = π and c2 = 1 for zero initial displacement and “triangular” initial velocity ut(x, 0) = 0.01x if 0 ≤ x ≤ 1/2π, ut(x, 0) = 0.01(π -
Name and explain the three kinds of boundary conditions for Laplace’s equation.
What happens to the frequency of an eigenfunction of a drum if you double the tension?
Verify (by substitution) that the given function is a solution of the PDE. Sketch or graph the solution as a surface in space.Laplace Equation (3)u = x/(x2 + y2), y/(x2 + y2)
Find the temperature in Prob. 11 with L = π, c = 1, and f (x) = 1 Data from Prob. 11 “Adiabatic” means no heat exchange with the neighborhood, because the bar is completely insulated, also at
Let f(x) = 1 when x > 0 and 0 when x
Find the deflection u(x, y, t) of the square membrane of side π and c2 = 1 for initial velocity 0 and initial deflection0.1 xy (π - x)(π - y)
Find the type, transform to normal form, and solve. Show your work in detail.uxx + 5uxy + 4uyy = 0
Find the electrostatic potential between two concentric spheres of radii r1 = 2 cm and r2 = 4 cm kept at the potentials U1 = 220 V and U2 = 140 V, respectively. Sketch and compare the
Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph u(x, t) as in Fig. 291 in
How do problems for the wave equation and the heat equation differ regarding additional conditions?

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