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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find a parametric representation and sketch the path.y = 1/x form (1, 1) to (5, 1/5)
Integrate counterclockwise or as indicated. Show the details. sin z C consists of the boundaries of the -dz, 4z2 – 8iz squares with vertices +3, ±3i counterclockwise and +1, ±i clockwise (see
Integrate. Show the details. Begin by sketching the contour. Why?
Find a parametric representation and sketch the path.Ellipse 4x2 + 9y2 = 36 counterclockwise
Integrate counterclockwise or as indicated. Show the details. tan z dz, C the boundary of the triangle with ±1 + 2i. vertices 0 and
Integrate. Show the details. Begin by sketching the contour. Why? dz, C consists of z - il = 3 counter- Jc z(z – 2i)² clockwise and |z| = 1 clockwise.
Integrate. Show the details. Begin by sketching the contour. Why? Ln (z + 3) dz, C the boundary of the square Jc (z – 2)(z + 1)? with vertices +1.5, +1.5i, counterclockwise.
Integrate counterclockwise or as indicated. Show the details. -dz, ze – 2iz C: |z| = 0.6
Find a parametric representation and sketch the path.Unit circle, clockwise
Find a parametric representation and sketch the path.From (0, 0) to (2, 1) along the axes
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = tan 1/4z
Find the path and sketch it.z(t) = 2 cos t + i sin t (0 ≤ t ≤ 2π)
Integrate. Show the details. Begin by sketching the contour. Why? 4z3 – 6 dz, C consists of |z| = 3 counter- 1 clockwise. z(z – 1 - ij2 clockwise and |z| = 1
Find the path and sketch it.z(t) = 5e-it (0 ≤ t ≤ 2)
Integrate. Show the details. Begin by sketching the contour. Why? z* + sin z dz, Jc (z – ij3 vertices +2, ±2i counterclockwise. C the boundary of the square with
Integrate the given function around the unit circle.(z2 sin z)/(4z - 1)
Verify Theorem 2 for the integral of ez from 0 to 1 + i(a) Over the shortest path. (b) Over the x-axis to 1 and then straight up to 1 + i.
Find the path and sketch it.z(t) = 1 + i + e-πit (0 ≤ t ≤ 2)
Integrate the given function around the unit circle.e2z/(πz - i)
Integrate counterclockwise around the unit circle. dz Jc (z – 21)°(z – i/2)?
If the integral of a function over the unit circle equals 2 and over the circle of radius 3 equals 6, can the function be analytic everywhere in the annulus1 < |z| < 3?
Find the path and sketch it.z(t) = t + (1 - t)2i (1 ≤ t ≤ 1)
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z + 5 - 5i| = 7
Integrate counterclockwise around the unit circle. e cos z dz Je (z - T/4)3
For what contours C will it follow from Theorem 1 that exp (1/z³) -dz = 0? z2 + 16 dz (a) 0, ,2 (b)
Find the path and sketch it.z(t) = 3 + i + (1 - i)t (0 ≤ t ≤ 3)
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z - 1 - i| = π/2
Integrate counterclockwise around the unit circle. dz c (2z – 1)®
Inequalities and equalityProve |Re z| ≤ |z|, |Im z| ≤ |z|.
Find the value of:sinh (1 + πi), sin (1 + πi)
Inequalities and equalityVerify (6) for z1 = 3 + i, z2 = -2 + 4i
Find the value of:Ln (0.6 + 0.8i)
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:a • (b - c), (a - b) • c
What curves are represented by the following? Sketch them.[t, 2, 1/t]
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 = x/(x2 + y2), P: (1, 1)
Find a parametric representationCircle in the yz-plane with center (4, 0) and passing through (0, 3). Sketch it.
Consider the flow with velocity vector v = xi. Show that the individual particles have the position vectors r(t) = c1eti + c2j + c3k with constant c1, c2, c3. Show that the particles that at t = 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, b × d, d × b, a × a
What kind of surfaces are the level surfaces f(x, y, z) = const?f = x - y2
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:4b × 3c + 12c × b
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.5(a × b) • c, a • (5b × c), (5a b c), 5(a • b) × c
Does div u = div v imply u = v or u = v + k (k constant)? Give reason.
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:3c - 6d, 3(c - 2d)
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 = (x2 + y2 + z2)-1/2, P: (12, 0, 16)
Sketch figures similar to Fig. 198. Try to interpret the field of v as a velocity field.v = -yi + xj 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.curl
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
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,
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,
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
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
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 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
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
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
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.

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