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Questions and Answers of Advanced Engineering Mathematics

Q:

Let A = [αjk] be an n × n matrix with (not necessarily distinct) eigenvalues λ1, · · · , λn. Show.The polynomial matrixhas the

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Find an eigenbasis and diagonalize.

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Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.

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Find the eigenvalues. Find the corresponding eigenvectors. 

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Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 × 3 consumption

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Find the eigenvalues. Find the eigenvectors.

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Is the matrix A Hermitian or skew-Hermitian? Find x̅TAx. Show the details.

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Summarize the main concepts and facts in this section, giving illustrative examples of your own.

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Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.

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x1C3H8 + x2O2 → x3CO2 + x4H2O means finding integer x1, x2, x3, x4 such that the numbers of atoms of carbon (C), hydrogen (H), and oxygen (O) are

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Find the Euclidean norm of the vectors:[1/2    1/3    -1/2    -1/3]T

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Showing all intermediate results, calculate the following expression or give reasons why they are undefined:BC, BCT, Bb, bTB

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Describe the region of integration and evaluate.

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What are typical applications of line integrals? Of surface integrals?

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Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [cosh y, -sinh x], R: 1 ≤ x ≤ 3,

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Verify (9) for f = 6y2, g = 2x2, S the unit cube in Prob. 3.Data from Prob. 3Verify (8) for f = 4y2, g = x2, S the surface of the “unit cube” 0

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Find the total mass of a mass distribution of density σ in a region T in space.σ = sin 2x cos 2y, T : 0 ≤ x ≤ 1/4π, 1/4π - x ≤ y ≤ 1/4π,

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What role did the gradient play in this chapter? The curl? State the definitions from memory.

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Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves

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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

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Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [x, y, z], S: r = [u cos v, u sin v, u2],0

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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

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Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [z2, 3/2x, 0], S: 0 ≤ x ≤ α, 0 ≤ y ≤ α, z =

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Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [x2 + y2, x2 - y2], R: 1 ≤ y ≤ 2

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What surface integrals can be converted to volume integrals? How?

Q:

Verify (8) for f = 4y2, g = x2, S the surface of the “unit cube” 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. What are the assumptions on f and g

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Describe the region of integration and evaluate.

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Find the total mass of a mass distribution of density σ in a region T in space.σ = e-x-y-z, T: 0 ≤ x ≤ 1 - y, 0 ≤ y ≤ 1, 0 ≤ z ≤ 2

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What line integrals can be converted to surface integrals and conversely? How?

Q:

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves

Q:

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

Q:

Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [0, x, 0], S: x2 + y2 + z2 = 1, x ≥ 0, y

Q:

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

Q:

Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [x2ey, y2ex], R the rectangle with

Q:

Find the total mass of a mass distribution of density σ in a region T in space.σ = xyz, T the box 0 ≤ x ≤ α, 0 ≤ y ≤ b, 0 ≤ z ≤ c

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What is path independence of a line integral? What is its physical meaning and importance?

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What is path independence of a line integral? What is its physical meaning and importance?

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Find the total mass of a mass distribution of density σ in a region T in space.σ = x2 + y2 + z2, T the box |x| ≤ 4, |y| ≤ 1, 0 ≤ z ≤ 2

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State from memory how to evaluate a line integral. A surface integral.

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Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves

Q:

Write a short report (1–2 pages) with examples on line integrals as generalizations of definite integrals. The latter give the area under a curve.

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Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [-x2, y2, 0], S: r = [u, v, 3u - 2v],

Q:

Make a list of the main ideas and facts on path independence and dependence in this section. Then work this list into a report. Explain the

Q:

Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [z2, -x2, 0], S the rectangle with vertices (0, 0, 0), (1,

Q:

Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [ y, -x], C the circle x2 + y2 = 1/4

Q:

Obtain κ and τ in Prob. 52 from (22*) and (23***) and the original representation in Prob. 54 with parameter t.Data from Prob. 52Show that the

Q:

Find the torsion of C: r(t) = [t, t2, t3], which looks similar to the curve in Fig. 212.

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Using (22*), show that for a curve y = f(x),

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Show that a circle of radius α has curvature 1/α.

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Find the speed of an artificial Earth satellite traveling at an altitude of 80 miles above Earth’s surface, where g = 31 ft/sec2. (The radius of

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Find the acceleration of the Earth toward the sun from (19) and the fact that Earth revolves about the sun in a nearly circular orbit with an almost

Q:

The use of a CAS may greatly facilitate the investigation of more complicated paths, as they occur in gear transmissions and other constructions. To

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Find the component of a in the direction of b. Make a sketch.When will the component (the projection) of a in the direction of b be equal to the

Q:

Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. FindDwf at P: (3, 0, 2)

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Find the forces in the system of two rods (truss) in the figure, where |p| = 1000 nt. Forces in equilibrium form a polygon, the force polygon.

Q:

Find the component of a in the direction of b. Make a sketch.a = [3, 4, 0], b = [4, -3, 2]

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Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Findgrad (div w)

Q:

Find the component of a in the direction of b. Make a sketch.a = [1, 1, 1], b = [2, 1, 3]

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Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of

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Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Finddiv (grad f), ∇2f, ∇2(xyf)

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Summarize the most important applications we have discussed in this section and give a few simple examples. No proofs.

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Same question as in Prob. 32 if |p| = 9, |q| = 6, |u| = 3.Data from Prob. 32If |p| = 6 and |q| = 4, what can you say about the magnitude and

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Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of

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Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Finddiv v, div w

Q:

Find the volume if the vertices are (1, 1, 1), (5, -7, 3), (7, 4, 8), and (10, 7, 4).

Q:

Show that Eq. (10) implies ℓ = ∫bα √1 + y'2 dx for the length of a plane curve C: y = f(x), z = 0, and α = x = b.

Q:

Find the volume if the edge vectors are i + j, -2i + 2k, and -2i - 3k. Make a sketch.

Q:

For what k is the resultant of [2, 0, -7], [1, 2, -3], and [0, 3, k] parallel to the xy-plane?

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Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of

Q:

Find the volume if the vertices are (0, 0, 0), (3, 1, 2), (2, 4, 0), (5, 4, 0).

Q:

Find the plane through (1, 3, 4), (1, -2, 6), and (4, 0, 7).

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Find the length and sketch the curve.r(t) = [α cos t, α sin t] from (α, 0) to (0, α)

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Find all v such that the resultant of v, p, q, u with p, q, u as in Prob. 21 is parallel to the xy-plane.Data from Prob. 21Find the resultant in

Q:

Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Find the angles if the vertices are (0, 0), (6, 0), (8, 3), and (2, 3).

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A force p = [4, 2, 0] is acting in a line through (2, 3, 0). Find its moment vector about the center (5, 1, 0) of a wheel.

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Find the area if the vertices are (0, 0, 1), (2, 0, 5), and (2, 3, 4).

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Find the length and sketch the curve.catenary r(t) = [t, cosh t] from t = 0 to t = 1.

Q:

What does it mean if |∇f(P)| > |∇f(Q)|at two points P and Q in a scalar field?

Q:

Find p such that u, v, w in Prob. 23 and p are in equilibrium.Data from Prob. 23u = [8, -1, 0], v = [1/2, 0, 4/3], w = [-17/2 , 1, 11/3]

Q:

Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:cos (α - β) = cosα cos β + sin α sin β. Obtain this by using a =

Q:

Find the component of v = [4, 7, 0] in the direction of w = [2, 2, 0]. Sketch it.

Q:

Find the area if the vertices are (4, 2, 0), (10, 4, 0), (5, 4, 0), and (11, 6, 0). Make a sketch.

Q:

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/t,

Q:

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

Q:

Find the work done by q = [5, 2, 0] in the displacement from (1, 1, 0) to (4, 3, 0).

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Find 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 sketch.

Q:

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) = [10 cos

Q:

Summarize the essential ideas and facts and give examples of your own.

Q:

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

Q:

Given the velocity potential of a flow, find the velocity v = ∇f of the field and its value v(P) at P. Sketch v(P) and the curve f = const passing

Q:

Find the resultant in terms of components and its magnitude.u = [8, -1, 0], v = [1/2, 0, 4/3], w = [-17/2 , 1, 11/3]

Q:

Find the angle between a and c. Between b and d. Sketch a and c.

Q:

Use your CAS to graph the following curves given in polar form ρ = ρ(θ), ρ2 = x2 + y2, tan θ = y/x, and investigate their form depending on

Q:

Prove (11)–(13). Give two typical examples for each formula.

Q:

Graph the following more complicated curves:(a) r(t) = [2 cos t + cos 2t, 2 sin t - sin 2t] (Steiner’s hypocycloid).(b) r (t) = [cos t + k cos 2t,

Q:

Given the velocity potential of a flow, find the velocity v = ∇f of the field and its value v(P) at P. Sketch v(P) and the curve f = const passing

Q:

Explain why setting t = -t* reverses the orientation of [α cos t, α sin t, 0].

Q:

Find the resultant in terms of components and its magnitude.p = [2, 3, 0], q = [0, 6, 1], u = [2, 0, -4]

Q:

Find the work done by a force p acting on a body if the body is displaced along the straight segment A̅B̅ from A to B. Sketch A̅B̅ and p. Show

Q:

Find u such that u and a, b, c, d above and u are in equilibrium.

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