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mathematics
advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

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)
Prove Eq. (7). Use Eq. (3) for |a+ b| and Eq. (6) to prove the square of Eq. (7), then take roots.
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.(1/|a|)a, (1/|b|)b, a • b/|b|, a • b/|a|
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = exyz
Find a parametric representationThe intersection of the circular cylinder of radius 1 about the z-axis and the plane z = y.
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:9/2 a - 3c, 9 (1/2 a - 1/3 c)
For what points P: (x, y, z) does ∇f with f = 25x2 + 9y2 + 16z2 have the direction from P to the origin?
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field.v = xi +yj P
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.div (u × v)
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 × b) × a, a × (b × a)
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|, |a| + |b|
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = z - √x2 + y2
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 = [-1, -2, 4], A: (0, 0, 0), B: (6, 7, 5)
Find a parametric representationHelix x2 + y2 = 25, z = 2 arctan (y/x).
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 = x2 - 6x - y2, P: (-1, 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:4a + 3b, -4a - 3b
Find a parametric representationIntersection of 2x - y + 3z = 2 and x + 2y - z = 3.
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 = [6, -3, -3], A: (1, 5, 2), B: (3, 4, 1)
When is u × v = v × u? When is u • v = v • u?
Calculate ∇2f by Eq. (3). Check by direct differentiation. Indicate when (3) is simpler. Show the details of your work.f = e2x cosh 2y
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 × b) × (c × d), (a b d)c - (a 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.div (grad ( fg))
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field.v = yi - xj P
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 = x(1 + (x2 + y2)-1), P: (1, 1)
Find the first and second derivatives of r = [3 cos 2t, 3 sin 2t, 4t].
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 - b c - b d - b), (a c d)
Find the most general v such that the resultant of v, a, b, c (see above) is parallel to the yz-plane.
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:a, b
Find the resultant in terms of components and its magnitude.p = [1, -2, 3], q = [3, 21, -16], u = [-4, -19, 13]
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 directed vertically upward?Data from Prob. 21Given the velocity potential of a flow, find the
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 an arrow.T = 3x2 - 2y2, P: (2.5, 1.8)
Find the first partial derivatives of v1 = [ex cos y, ex sin y] and v2 = [cos x cosh y, -sinx sinh y].
Prove (13)–(16), which are often useful in practical work, and illustrate each formula with two examples. For (13) choose Cartesian coordinates such that d = [d1, 0, 0] and c = [c1, c2, 0]. Show that each side of (13) then equals [-b2c2d1, b1c2d1, 0], and give reasons why the two sides are then
Find the angle between the two planes P1 : 4x - y + 3z = 12 and P2: x + 2y + 4z = 4. Make a sketch.
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:a + c, b + c
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work. 5s + 1 s2 – 25
Using Theorem 3, find f (t) if L(F) equals: 3s + 4 s* + k²s² .2
What form does a 3 X 3 matrix have if it is symmetric as well as skew-symmetric?
Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). sin 20 cos 20 cos 20 -sin 20
Show that the representation v = c1a(1) + . . . + cna(n) of any given vector in an n-dimensional vector space V in terms of a given basis a(1), . . ., a(n) for V is unique. Take two representations and consider the difference.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose. a
All skew-symmetric 3 x 3 matrices.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose. -4 -4 2 2 6.
Solve the linear system given explicitly or by its augmented matrix. Show details. 16 4 -8 -5 -21 -1 3 -6 1 3. 2.
If U1, U2 are upper triangular and L1, L2 are lower triangular, which of the following are triangular?U1 + U2, U1U2, U12, U1 + L1, U1L1, L1 + L2
Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). -4 8. 13 3
All functions y(x) = α cos 2x + b sin 2x with arbitrary constants α and b.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose. -1 -4 4
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.2A + 4B, 4B + 2A, 0A + B, 0.4B - 4.2A Let 4 A 6. B = 3 -3 -2 4 -2 2 -4 D =| -2 5 1 -1 2. 27 E = 3 4 1.5 3 w =-30 -3.0 10
Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). Г1 3 4 5 6. [7 2.
All n x n matrices A with fixed n and det A = 0.
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose. 8 16 4 16 8. 8 16 4 2 16 8. 4 2. 2. 4,
Find the rank. Find a basis for the row space. Find a basis for the column space. Row-reduce the matrix and its transpose. -2 1 -2 -4 1 -4 -11
All 3 x 2 matrices [αjk] with first column any multiple of [3 0 -5]T.
Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). calon calon
Solve the linear system given explicitly or by its augmented matrix. Show details. 01 -2 -3 -6 15 1 1 -1 2. 3. 2.
Showing all intermediate results, calculate the following expression or give reasons why they are undefined:AAT, A2, BBT, B2 Let 4 -2 3 -3 A =-2 1 B = -3 6. 1 2 2 -2 1. a = [1 -2 0], 0), b = 3 2 -2
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.(C + D) + E, (D + E) + C, 0(C - E) + 4D, A - 0C Let 4 A 6. B = 3 -3 -2 4 -2 2 -4 D =| -2 5 1 -1 2. 27 E = 3 4 1.5 3 w =-30 -3.0 10
Show the following:Rank BTAT = rank AB. (Note the order!)
Showing the details, evaluate: c| b.
Find the inverse transformation. Show the details.y1 = 3x1 + 2x2y2 = 4x1 + x2
Solve the linear system given explicitly or by its augmented matrix. Show details. Г2 3 1 -11 -2 -4 -1 3 -3 3 3 4 -7 -7
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.(5u + 5v) - 1/2 w, -20(u + v) + 2w, E - (u + v), 10(u + v) + w Let 4 A 6. B = 3 -3 -2 4 -2 2 -4 D =| -2 5 1 -1 2. 27 E = 3 4 1.5 3 w =-30 -3.0 10
Find the inverse transformation. Show the details.y1 = 0.2x1 - 0.1x2y2 = -0.2x2 + 0.1x3y3 = 0.1x1 + 0.1x3
Showing the details, calculate the following expressions or give reason why they are not defined, whenA-1, B-1 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.15v - 3w - 0u, -3w + 15v, D - u + 3C, 8.5w - 11.1u + 0.4v Let 4 A 6. B = 3 -3 -2 4 -2 2 -4 D =| -2 5 1 -1 2. 27 E = 3 4 1.5 3 w =-30 -3.0 10
Find the value of the determinant of the n x n matrix An with main diagonal entries all 0 and all others 1. Try to find a formula for this. Try to prove it by induction. Interpret A3 and A4 as incidence matrices (as in Problem Set 7.1 but without the minuses) of a triangle and a tetrahedron,
Find the rank by Theorem 3 (which is not very practical) and check by row reduction. Show details. 4 -6 4 10 -9- 10
Showing the details, calculate the following expressions or give reason why they are not defined, when(A + AT)(B - BT) 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Formula (4) is occasionally needed in theory. To understand it, apply it and check the result by Gauss€“Jordan:In Prob. 6Data from Prob. 6Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). -4 8. 13 3
Determine the equilibrium solution (D1 = S1, D2 = S2) of the two-commodity market with linear model (D, S, P = demand, supply, price; index 1 = first commodity, index 2 = second commodity)D1 = 40 - 2P1 - P2, S1 = 4P1 - P2 + 4,D2 = 5P1 - 2P2 + 16, S2 = 3P2 - 4.
Showing the details, find all solutions or indicate that no solution exists.5x - 3y + z = 72x + 3y - z = 08x + 9y - 3z = 2
Are the following sets of vectors linearly independent? Show the details of your work. [0.4 -0.2 0.2], [O 0 0], [3.0 -0.6 1.5]
Find all vectors in R3 orthogonal to [2 0 1]. Do they form a vector space?
Solve by Cramer’s rule. Check by Gauss elimination and back substitution. Show details. 3x - 2y + z = 13-2x + y + 4z = 11 x + 4y - 5z = -31
Showing the details, find all solutions or indicate that no solution exists.-6x + 39y - 9z = -122x - 13y + 3z = 4
The idea is that elementary operations can be accomplished by matrix multiplication. If A is an m x n matrix on which we want to do an elementary operation, then there is a matrix E such that EA is the new matrix after the operation. Such an E is called an elementary matrix. This idea can be
Find all 2 X 2 matrices A = [αjk] that commute with B = [bjk], where bjk = j + k.
In a production process, let N mean “no trouble” and T “trouble.” Let the transition probabilities from one day to the next be 0.8 for N → N, hence 0.2 for N → T, and 0.5 for T → N, hence 0.5 for T → T. If today there is no trouble, what is the probability of N two days after today?
Beginning with the last of the vectors [3 0 1 2], [6 1 0 0], [12 1 2 4], [6 0 2 4], and [9 0 1 2], omit
Showing the details, find all solutions or indicate that no solution exists. -8x + 2z = 1 6y + 4z = 312x + 2y = 2
In a community of 100,000 adults, subscribers to a concert series tend to renew their subscription with probability 90% and persons presently not subscribing will subscribe for the next season with probability 0.2%. If the present number of subscribers is 1200, can one predict an increase,
Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, v2, · · · denote components.)All vectors in R3 with 3v2 + v3 = k
Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, v2, · · · denote components.)All vectors in Rn with the first n - 2 components zero.
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. [2.0 0.4] 2.0] [0.4
Find the eigenvalues. Find the corresponding eigenvectors.
Verify this for A and A = P-1AP. If y is an eigenvector of P, show that x = Py are eigenvectors of A. Show the details of your work. 7 -5 P = 10 %3D 2 -7 -1
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. -b
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors. -1 + i
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors.
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. cos 0 -sin e sin 0 os e cos
Find the eigenvalues. Find the corresponding eigenvectors. 4
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. [5 13
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. 1.25 0.75 0.75 1.25
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. Га a
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors. 2 + 2i 2 - 2i 2 + 2i 2 - 2i
Find the limit state of the Markov process modeled by the given matrix. Show the details. [0.4 0.3 0.3 0.1 0.3 0.6 0.3 3 0.1 0.6
Find the eigenvalues. Find the corresponding eigenvectors. a -b a
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. cos e - sin e cos e sin e
Find the growth rate in the Leslie model (see Example 3) with the matrix as given. Show the details. 9.0 5.0 0.4 0.4

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