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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Summarize the essential ideas and facts and give examples of your own.
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
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 through P.At what points is the flow in Prob. 21
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]
Find the angle between a and c. Between b and d. Sketch a and c.
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 parameters α and b.
Prove (11)–(13). Give two typical examples for each formula.
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, sin t - k sin 2t] with k = 10, 2, 1, 1/2, 0, -1/2,
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 through P.f = ex cos y, P: (1, 1/2π)
Explain why setting t = -t* reverses the orientation of [α cos t, α sin t, 0].
Find the resultant in terms of components and its magnitude.p = [2, 3, 0], q = [0, 6, 1], u = [2, 0, -4]
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 the details.Is the work done by the resultant of two
Find u such that u and a, b, c, d above and u are in equilibrium.
Plot by arrows:(a) v = [x, x2] (b) v = [1/y, 1/x](c) v = [cos x, sin x] (d) v = e-(x2+y2) [x, -y]
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 through P.f = cos x cosh y, P: (1/2π, ln 2)
What laws do Probs. 12–16 illustrate?Data from Prob. 12(a + b) + c, a + (b + c)Data from Prob. 13b + c, c + bData from Prob. 143c - 6d, 3(c - 2d)Data from Prob. 157(c - b), 7c - 7bData from Prob.
Find a parametric representationHyperbola 4x2 - 3y2 = 4, z = -2.
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = 1/(x2 + y2 + z2)
Let a = [4, 7, 0], b = [3, -1, 5], c = [-6, 2, 0], and d = [1, -2, 8]. Calculate the following expressions. Try to make a sketch.a × b - b × a, (a × c) • c, |a × b|
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:(i j k), (i k j)
With respect to right-handed coordinates, let u = [y, z, x], v = [yz, zx, xy], f = xyz, and g = x + y + z. Find the given expressions. Check your result by a formula in Proj. 14 if applicable.curl
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field. v = xi - yj
Same question as in Prob. 16 when f = 25x2 + 4y2.Data from Prob. 16For what points P: (x, y, z) does ∇f with f = 25x2 + 9y2 + 16z2 have the direction from P to the origin?
Let a = [3, 2, 0] = 3i + 2j; b = [-4, 6, 0] = 4i + 6j, c = [5, -1, 8] = 5i - j + 8k, d = [0, 0, 4] = 4k.Find:(7 - 3) a, 7a - 3a
Find a parametric representationCircle 1/2x2 + y2 = 1, z = y.
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 the details.p = [2, 5, 0], A: (1, 3, 3), B: (3, 5, 5)
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = ln (x2 + y2)
Let a = [4, 7, 0], b = [3, -1, 5], c = [-6, 2, 0], and d = [1, -2, 8]. Calculate the following expressions. Try to make a sketch.(a b d), (b a d), (b d a)
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:(b × c) × d, b × (c × d)
With respect to right-handed coordinates, let u = [y, z, x], v = [yz, zx, xy], f = xyz, and g = x + y + z. Find the given expressions. Check your result by a formula in Proj. 14 if applicable.v •
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field. v = xj
The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P.f = 4x2 + 9y2 + z2, P: (5, -1, -11)
Let a = [3, 2, 0] = 3i + 2j; b = [-4, 6, 0] = 4i + 6j, c = [5, -1, 8] = 5i - j + 8k, d = [0, 0, 4] = 4k.Find:7(c - b), 7c - 7b
Find a parametric representationStraight line y = 4x - 1, z = 5x.
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = cos2 x + sin2 y
Let a = [4, 7, 0], b = [3, -1, 5], c = [-6, 2, 0], and d = [1, -2, 8]. Calculate the following expressions. Try to make a sketch.6(a × b) × d, a × 6(b × d), 2a × 3b × d
Prove the parallelogram equality. Explain its name.
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:(a + d) × (d + a)
With respect to right-handed coordinates, let u = [y, z, x], v = [yz, zx, xy], f = xyz, and g = x + y + z. Find the given expressions. Check your result by a formula in Proj. 14 if applicable.curl (u
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field. v = i + j
Verify the Cauchy–Schwarz and triangle inequalities for the above a and b.
The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P.f = ln (x2 + y2), P: (8, 6)
Let a = [3, 2, 0] = 3i + 2j; b = [-4, 6, 0] = 4i + 6j, c = [5, -1, 8] = 5i - j + 8k, d = [0, 0, 4] = 4k.Find:b + c, c + b
Find a parametric representationStraight line through (2, 1, 3) in the direction of i + 2j.
The velocity vector v(x, y, z) of an incompressible fluid rotating in a cylindrical vessel is of the form v = w × r, where w is the (constant) rotation vector. Show that div v = 0.
Let a = [4, 7, 0], b = [3, -1, 5], c = [-6, 2, 0], and d = [1, -2, 8]. Calculate the following expressions. Try to make a sketch.b × c, c × b, c × c, c • c
Prove the Cauchy–Schwarz inequality.
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:c × (a + b), a × c + b × c
What kind of surfaces are the level surfaces f(x, y, z) = const?f = z - (√x2 + y2)
Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles).v = [x, y, -z]
Let a = [3, 2, 0] = 3i + 2j; b = [-4, 6, 0] = 4i + 6j, c = [5, -1, 8] = 5i - j + 8k, d = [0, 0, 4] = 4k.Find:(a + b) + c, a + (b + c)
With respect to right-handed Cartesian coordinates, let a = [2, 1, 0], b = [-3, 2, 0] c = [1, 4, -2], and d = [5, -1, 3]. Showing details, find:3c × 5d, 15d × c, 15d • c, 15c • d
The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P.f = xy, P: (-4, 5)
Let a = [3, 2, 0] = 3i + 2j; b = [-4, 6, 0] = 4i + 6j, c = [5, -1, 8] = 5i - j + 8k, d = [0, 0, 4] = 4k.Find:2a, 1/2a, -a
Find a parametric representationCircle in the plane z = 1 with center (3, 2) and passing through the origin.
Show that the flow with velocity vector v = yi is incompressible. Show that the particles that at time t = 0 are in the cube whose faces are portions of the planes x = 0, x = 1, y = 0, y = 1, z = 0,
Let a = [4, 7, 0], b = [3, -1, 5], c = [-6, 2, 0], and d = [1, -2, 8]. Calculate the following expressions. Try to make a sketch.a • c, 3b • 8d, 24d • b, a • a
What laws do Probs. 1 and 4–7 illustrate?Data from Prob. 1a • b, b • a, b • cData from Prob. 4|a + b|, |a| + |b|Data from Prob. 5Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|b +
What kind of surfaces are the level surfaces f(x, y, z) = const?f = 5x2 + 2y2
Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles).v = [ y, -2x, 0]
Granted sufficient differentiability, which of the following expressions make sense? f curl v, v curl f,u ×v, u × v × w, f • v, f • (v × w), u • (v × w),v × curl v, div (fv), curl (fv),
Summarize the most important applications discussed in this section. Give examples. No proofs.
Prove and illustrate by an example.∇(f/g) = (1/g2)(g∇f - f∇g)
Find the terminal point Q of the vector v with components as given and initial point P. Find |v|.6, 1, -4; P: (-6, -1, -4)
What curves are represented by the following? Sketch them.[cos t, sin 2t, 0]
Write down the definitions and explain the significance of grad, div, and curl.
Useful Formulas for the Divergence. Prove(a) div (kv) = k div v (k constant)(b) div (fv) = f div v + v • ∇f(c) div (f∇g) = f ∇2g + ∇f • ∇g(d) div (f∇g) - div (g∇f) = f∇2g -
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:15a • b + 15a • c, 15a • (b + c)
What does (a b c) = 0 imply with respect to these vectors?
What kind of surfaces are the level surfaces f(x, y, z) = const?f = 4x - 3y + 2z
Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles).v = [0, 3z2, 0]
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:5a • 13b, 65a • b
Prove and illustrate by an example.∇(fn) = nfn-1 ∇f
Find the terminal point Q of the vector v with components as given and initial point P. Find |v|.1/2, 3, -1/4; P: (7/2, -3, 3/4)
What curves are represented by the following? Sketch them.[4 cos t, 4 sin t, 3t]
Can a moving body have constant speed but variable velocity? Nonzero acceleration?
For what v3 is v = [ex cos y, ex sin y, v3] solenoidal?
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|a • c|, |a||c|
A wheel is rotating about the y-axis with angular speed ω = 20 sec-1. The rotation appears clockwise if one looks from the origin in the positive y-direction. Find the velocity and speed at the
Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work.v = [0, 0, e-x sin y]
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = 9x2 + 4y2
If r(t) represents a motion, what are r'(t), |r'(t)|, r"(t), and |r"(t)|?
Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.f = x4 + y4
Find the components of the vector v with initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector u in the direction of v.P: (0, 0, 0), Q: (2, 1, -2)
What curves are represented by the following? Sketch them.[2 + 4 cos t, 1 + sin t, 0]
How is the derivative of a vector function defined? What is its significance in geometry and mechanics?
Find div v and its value at P.v = x2y2z2[x, y, z], P: (3, -1, 4)
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|b + c|, |b| + |c|
Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work.v = xyz [x, y, z]
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = y/(x2 + y2)
When is a vector product the zero vector? What is orthogonality?
Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.f = y/x
Find the components of the vector v with initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector u in the direction of v.P: (-3.0, 4,0, -0.5), Q: (5.5, 0, 1.2)
What curves are represented by the following? Sketch them.[0, t, t3]
What are right-handed and left-handed coordinates? When is this distinction important?
Find div v and its value at P.v = (x2 + y2)-1[x, y]
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|a|, |2b|, |-c|
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = 3x - 4y
What is an inner product, a vector product, a scalar triple product? What applications motivate these products?
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:(-3a + 5c) • b, 15(a - c) • b

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