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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Let f(z) be analytic at z0. Suppose that f'(z0) = 0, · · ·, f(k-1)(z0) = 0. Then the mapping w = f(z) magnifies angles with vertex at z0 by a factor k. Illustrate this with examples for k =
Experiment with integrals ∫∞ -∞ f(x) dx, f(x) = [(x - α1)(x - α2) · · · (x - αk)]-1, αj real and all different, k > 1. Conjecture that the principal value of these
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = 1/2z2
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = 1/z
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = ez
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = Ln z
What is a conformal mapping? Why does it occur in complex analysis?
At what points are w = z5 - z and w = cos (πz2) not conformal?
What happens to angles at z0 under a mapping w = f(z) if f'(z0) = 0, f"(z0) = 0, f"'(z0) ≠ 0?
What is a linear fractional transformation? What can you do with it? List special cases.
What is the extended complex plane? Ways of introducing it?
What is a fixed point of a mapping? Its role in this chapter? Give examples.
To get a feel for increase in accuracy, integrate x2 from 0 to 1 by (2) with h = 1, 0.5, 0.25, 0.1.
Let A = [αjk] be an n × n matrix with (not necessarily distinct) eigenvalues λ1, · · · , λn. Show. The polynomial matrix has the eigenvalues where j = 1,· · ·,n, and the same eigenvectors
Find an eigenbasis and diagonalize.
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.
Find the eigenvalues. Find the corresponding eigenvectors.
Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 × 3 consumption matrix where αjk is the fraction of the output of industry
Find the eigenvalues. Find the eigenvectors.
Is the matrix A Hermitian or skew-Hermitian? Find x̅TAx. Show the details.
Summarize the main concepts and facts in this section, giving illustrative examples of your own.
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details.
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 the same on both sides of this reaction, in which
Find the Euclidean norm of the vectors:[1/2 1/3 -1/2 -1/3]T
Showing all intermediate results, calculate the following expression or give reasons why they are undefined:BC, BCT, Bb, bTB Let A C = 4-2 -2 1 0 3 -2 1 2 1 2 3 6 2 B || 1 -3 -3 1 00 0 -2 a = [1 -2
Describe the region of integration and evaluate.
What are typical applications of line integrals? Of surface integrals?
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, x ≤ y ≤ 3x
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 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. What are
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, 0 ≤ z ≤ 6
What role did the gradient play in this chapter? The curl? State the definitions from memory.
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
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 as in Prob. 4, C the quarter-circle
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 ≤ u ≤ 4, -π ≤ v ≥ π
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.
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 = 1
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 - x2
What surface integrals can be converted to volume integrals? How?
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 in (8)? Must f and g be harmonic?
Describe the region of integration and evaluate.
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
What line integrals can be converted to surface integrals and conversely? How?
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
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 as in Prob. 2, C from (0, 0) straight
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 ≥ 0, z ≥ 0
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.
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 vertices (0, 0), (2, 0), (2, 3), (0, 3)
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
What is path independence of a line integral? What is its physical meaning and importance?
What is path independence of a line integral? What is its physical meaning and importance?
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
State from memory how to evaluate a line integral. A surface integral.
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
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. Explain the corresponding geometric interpretation
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], 0 ≤ u ≤ 1.5, -2 ≤ v ≤ 2
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 definitions and the practical usefulness of the theorems,
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, 0, 0), (0, 4, 4), (1, 4, 4)
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
Obtain κ and τ in Prob. 52 from (22*) and (23***) and the original representation in Prob. 54 with parameter t. Data from Prob. 52 Show that the helix [α cos t, α sin t, ct] can be represented by
Find the torsion of C: r(t) = [t, t2, t3], which looks similar to the curve in Fig. 212.
Using (22*), show that for a curve y = f(x),
Show that a circle of radius α has curvature 1/α.
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 the Earth is 3960 miles.)
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 constant speed of 30 km/s.
The use of a CAS may greatly facilitate the investigation of more complicated paths, as they occur in gear transmissions and other constructions. To grasp the idea, using a CAS, graph the path and
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 component (the projection) of b in the direction of a?
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. FindDwf at P: (3, 0, 2)
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.
Find the component of a in the direction of b. Make a sketch.a = [3, 4, 0], b = [4, -3, 2]
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Findgrad (div w)
Find the component of a in the direction of b. Make a sketch.a = [1, 1, 1], b = [2, 1, 3]
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.When will the diagonals
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Finddiv (grad f), ∇2f, ∇2(xyf)
Summarize the most important applications we have discussed in this section and give a few simple examples. No proofs.
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 direction of the resultant? Can you think of an
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.Find all unit vectors a
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Finddiv v, div w
Find the volume if the vertices are (1, 1, 1), (5, -7, 3), (7, 4, 8), and (10, 7, 4).
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.
Find the volume if the edge vectors are i + j, -2i + 2k, and -2i - 3k. Make a sketch.
For what k is the resultant of [2, 0, -7], [1, 2, -3], and [0, 3, k] parallel to the xy-plane?
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.For what values of α1
Find the volume if the vertices are (0, 0, 0), (3, 1, 2), (2, 4, 0), (5, 4, 0).
Find the plane through (1, 3, 4), (1, -2, 6), and (4, 0, 7).
Find the length and sketch the curve.r(t) = [α cos t, α sin t] from (α, 0) to (0, α)
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 terms of components and its magnitude.p = [2, 3, 0], 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).
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.
Find the area if the vertices are (0, 0, 1), (2, 0, 5), and (2, 3, 4).
Find the length and sketch the curve.catenary r(t) = [t, cosh t] from t = 0 to t = 1.
What does it mean if |∇f(P)| > |∇f(Q)|at two points P and Q in a scalar field?
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]
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 = [cos α, sin α], b = [cos β, sin β] where 0 ≤
Find the component of v = [4, 7, 0] in the direction of w = [2, 2, 0]. Sketch it.
Find the area if the vertices are (4, 2, 0), (10, 4, 0), (5, 4, 0), and (11, 6, 0). Make a sketch.
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, 0], P: (2, 1/2, 0)
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 the given point P. Sketch that direction at P as
Find the work done by q = [5, 2, 0] in the displacement from (1, 1, 0) to (4, 3, 0).
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.
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 t, 1, 10 sin t], P: (6, 1, 8)

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