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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

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 = cos y sinh x, sin y cosh x
Why and where did the error function occur?
Find the steady-state temperature in a semicircular thin plate r < α, 0 < θ < π with the semicircle r = α kept at constant temperature u0 and the segment -α < x < α at 0.
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 = arctan (y/x)
“Adiabatic” means no heat exchange with the neighborhood, because the bar is completely insulated, also at the ends. Physical Information: The heat flux at the ends is proportional to the value
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 the type, transform to normal form, and solve. Show your work in detail.uxx + 2uxy + uyy = 0
Substituting u(r) with r as in Prob. 9 into uxx + uyy + uzz = 0, verify that u" + 2u'/r = 0, in agreement with (7).Data from Prob. 9Show that the only solution of Laplace’s equation depending only
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
Explain mathematically (not physically) why we got exponential functions in separating the heat equation, but not for the wave equation.
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 sin 2x sin 4y
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 → ∞ for
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 = ex cos y, ex sin y
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 (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-π2t cos 25x
When did we use polar coordinates? Cylindrical coordinates? Spherical coordinates?
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
In Prob. 8 find the temperature at any time.Data from Prob. 8If 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)
Guess what the equipotential lines u(r, θ) = const in Probs. 5 and 7 may look like. Then graph some of them, using partial sums of the series.Data from Prob. 5The electrostatic potential satisfies
Find the type, transform to normal form, and solve. Show your work in detail.uxx + 4uyy = 0
Show that the only solution of Laplace’s equation depending only on r = √x2 + y2 + z2 is u = c/r + k with constant c and k.
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
What do you remember about types of PDEs? Normal forms? Why is this important?
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 → ∞ for
(a) Write a program that gives and graphs partial sums of (15). Apply it to Probs. 5 and 6. Do the graphs show that those partial sums satisfy the boundary condition (3a)? Explain why. Why is the
Using (13) sketch or graph a figure similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string (length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial
Verify that the potential u = c/r, r = √x2 + y2 + z2 satisfies Laplace’s equation in spherical coordinates.
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
What are the eigenfunctions and their frequencies of the vibrating string? Of the vibrating membrane?
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-ω2c2t cos ωx
Find the temperature u(x, t) in a bar of silver of length 10 cm and constant cross section of area 1 cm2 (density 10.6 g/cm3, thermal conductivity 1.04 cal/(cm sec ºC), specific heat 0.056
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref (x) = (sin x)/x.
Using (13) sketch or graph a figure similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string (length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial
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
When and why did Legendre’s equation occur? Bessel’s equation?
Solve by Laplace Transforms Solve Prob. 5 by separating variables Data from Prob. 5
Represent f (x, y) by a series (15), wheref (x, y) = xy, α and b arbitrary
Find the temperature u(x, t) in a bar of silver of length 10 cm and constant cross section of area 1 cm2 (density 10.6 g/cm3, thermal conductivity 1.04 cal/(cm sec ºC), specific heat 0.056
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref(x) = x if |x| < 1 and 0 otherwise
Using (13) sketch or graph a figure similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string (length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial
Study the Gibbs phenomenon in Example 1 (Fig. 314) graphically.
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
What role did Fourier series play in this chapter? Fourier integrals?
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.Wave Equation (1) with suitable cu = sin αt sin bx
Find the temperature u(x, t) in a bar of silver of length 10 cm and constant cross section of area 1 cm2 (density 10.6 g/cm3, thermal conductivity 1.04 cal/(cm sec ºC), specific heat 0.056
Using (6), obtain the solution of (1) in integral form satisfying the initial condition u (x, 0) = f(x), wheref (x) = |x| if |x| < 1 and 0 otherwise
Using (13) sketch or graph a figure similar to Fig. 291 in Sec. 12.3) of the deflection u(x, t) of a vibrating string (length L = 1, ends fixed, c = 1) starting with initial velocity 0 and initial
Determine and sketch or graph the sets in the complex plane given by|z + 1 - 5i| ≤ 3/2
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
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [0, sin y, cos z], S the cylinder x = y2, where 0 ≤ y ≤ π/4 and 0 ≤ z ≤ y
Calculate ∫C F(r) • dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details.F = [x2, y2, z2], C: r = [cos t, sin t,
Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves u = const and v = const) of the surface and a
Where did orientation of a surface play a role? Explain.
Describe the region of integration and evaluate. Prob. 6, order reversed. Data from Prob. 6
Find the total mass of a mass distribution of density σ in a region T in space.σ = arctan (y/x), T: x2 + y2 + z2 ≤ α2, z ≥ 0
Use the divergence theorem, assuming that the assumptions on T and S are satisfied. Show that a region T with boundary surface S has the volume
State the divergence theorem from memory. Give applications.
Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [ey/x, ey ln x + 2x], R: 1 + x4 ≤ y ≤ 2
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.Verify Stokes’s theorem for F and S in Prob. 5.Data from Prob. 5F = [z2, 3/2x, 0], S: 0 ≤ x ≤ α, 0 ≤ y
Show that the form under the integral sign is exact in the plane (Probs. 3–4) or in space (Probs. 5–9) and evaluate the integral. Show the details of your work.
Calculate ∫C F(r) • dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details.F = [x + y, y + z, z + x], C: r = [2t,
In some line and surface integrals we started from vector functions, but integrands were scalar functions. How and why did we proceed in this way?
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [0, sinh z, cosh x], S: x2 + z2 = 4, 0 ≤ x ≤ 1/√2, 0 ≤ y ≥ 5, z ≥ 0
Find the volume of the given region in space.The region beneath z = 4x2 + 9y2 and above the rectangle with vertices (0, 0), (3, 0), (3, 2), (0, 2) in the xy-plane.
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [x2, 0, z2], S the surface of the box |x| ≤ 1, |y| ≤ 3, 0 ≤ z ≤ 2
Use the divergence theorem, assuming that the assumptions on T and S are satisfied. Find the volume under a hemisphere of radius a from in Prob. 7. Data from Prob. 7 Show that a region T with
State Laplace’s equation. Explain its physical importance. Summarize our discussion on harmonic functions.
Find the volume of the given region in space.The first octant region bounded by the coordinate planes and the surfaces y = 1 - x2, z = 1 - x2. Sketch it.
Apply (4) to figures of your choice whose area can also be obtained by another method and compare the results.
Evaluate ∫C F • r' ds, F = (x2 + y2)-1[-y, x], C: x2 + y2 = 1, z = 0, oriented clockwise. Why can Stokes’s theorem not be applied? What (false) result would it give?
Write a program for evaluating surface integrals (3) that prints intermediate results (F, F • N, the integral over one of the two variables). Can you obtain experimentally some rules on functions
Calculate ∫C F(r) • dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details.F = [e-x, e-y, e-z], C: r = [t, t2, t]
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 the volume of the given region in space.The region above the xy-plane and below the paraboloid z = 1 - (x2 + y2).
Represent the paraboloid in Prob. 5 so that N∼(0, 0) ≠ 0 and show N∼.Data from Prob. 5Familiarize yourself with parametric representations of important surfaces by deriving a
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [ex, ey, ez], S the surface of the cube |x| ≤ 1, |y| ≤ 1, |z| ≤ 1
Use the divergence theorem, assuming that the assumptions on T and S are satisfied. Find the volume of a ball of radius α from Prob. 10. Data from Prob. 10 Show that a region T with boundary surface
Make a list of ideas and results on this topic in this chapter. See whether you can rearrange or combine parts of your material. Then subdivide the material into 3–5 portions and work out the
Using (9), find the value of ∫C ∂w/∂n ds taken counterclockwise over the boundary C of the region R.w = cosh x, R the triangle with vertices (0, 0), (4, 2), (0, 2).
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 = [-5y, 4x,
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).2ex2 (x cos 2y dx - sin 2y dy)
Evaluate these integrals for the following data. Indicate the kind of surface. Show the details.G = x + y + z, z = x + 2y, 0 ≤ x ≤ π, 0 ≤ y ≤ x
Let F be a vector function defined on a curve C. Let |F| be bounded, say |F| ≤ M on C, where M is some positive number. Show that
Find the center of gravity (x̅, y̅) of a mass of density f(x, y) = 1 in the given region R.
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
z = f(x, y). Show that z = f(x, y) or g = z - f(x, y) = 0 can be written (fu = ∂f/∂u, etc.)
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [sin y, cos x, cos z], S, the surface of x2 + y2 ≤ 4, |z| ≤ 2 (a cylinder and two disks!)
Using (9), find the value of ∫C ∂w/∂n ds taken counterclockwise over the boundary C of the region R.w = ex cos y + xy3, R: 1 ≤ y ≤ 10 - x2, x ≥ 0
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 = [y2, x2,
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).x2y dx - 4xy2 dy + 8z2x dz
Evaluate these integrals for the following data. Indicate the kind of surface. Show the details.G = (1 + 9xz)3/2, S: r = [u, v, u3], 0 ≤ u ≤ 1, -2 ≤ v ≤ 2
Evaluate them with F or f and C as follows.F = [y2, z2, x2], C: r = [3 cos t, 3 sin t, 2t], 0 ≤ t ≤ 4π
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 the center of gravity (x̅, y̅) of a mass of density f(x, y) = 1 in the given region R.
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Cylinder of revolution (x - 2)2 + (y + 1)2 = 25
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [2x2, 1/2y2, sin πz], S the surface of the tetrahedron with vertices (0, 0, 0), (1, 0, 0),
Using (9), find the value of ∫C ∂w/∂n ds taken counterclockwise over the boundary C of the region R.w = x3 - y3, 0 ≤ y ≤ x2, |x| ≤ 2

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