New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
mathematics
advanced engineering mathematics
Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.Verify Stokes’s theorem for F and S in Prob. 6.Data from Prob. 6F = [y3, -x3, 0], S: x2 + y2 ≤ 1, z = 0
Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [x2y2, -x/y2], R: 1 ≤ x2 + y2 ≤ 4, x ≥ 0, y ≥ x. Sketch R.
Use the divergence theorem, assuming that the assumptions on T and S are satisfied.Using the third expression for v in Prob. 7, V = πα2 h/3 verify for the volume of a circular cone of height h and radius of base α.Data from Prob. 7Show that a region T
Find the total mass of a mass distribution of density σ in a region T in space.σ = x2 + y2, T as in Prob. 7Data from Prob. 7σ = arctan (y/x), T: x2 + y2 + z2 ≤ α2, z ≥ 0
Describe the region of integration and evaluate. T/4 (cos y x² sin y dx dy 0.
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 normal vector N = ru × rv of the surface. Show the details of your work.r(u, v) = [αu cosh v, bu
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 = [ex, cosh y, sinh z], C: r = [t, t2, t3] from (0, 0, 0) to (1/2, 1/4, 1/8). Sketch C.
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [tan xy, x, y], S: y2 + z2 = 1, 2 ≤ x ≤ 5, 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 the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [z2, x2, y2], S: z = √x2 + y2, y ≥ 0, 0 ≤ z ≤ h
Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [-e-x cos y, -e-x sin y], R the semidisk x2 + y2 ≤ 16, x ≥ 0
Verify (9) for f = x2, g = y4, 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 the assumptions on f and g in (8)? Must f and g be harmonic?
Find the total mass of a mass distribution of density σ in a region T in space.σ = x2y2z2, T the cylindrical region x2 + z2 ≤ 16, |y| ≤ 4
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 normal vector N = ru × rv of the surface. Show the details of your work.r(u, v) = [u cos v, u
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 = [x - y, y - z, z - x], C: r = [2 cos t, t, 2 sin t] from (2, 0, 0) to (2, 2π,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. (1, 1,0) e + +x dx + y dy + z dz) (0,0, 0)
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [y3, -x3, 0], S: x2 + y2 ≤ 1, z = 0
Describe the region of integration and evaluate.Prob. 3, order reversed.Data from Prob. 3 .3 y (x2 + y²) dx dy
Find the total mass of a mass distribution of density σ in a region T in space.σ as in Prob. 3, T the tetrahedron with vertices (0, 0, 0), (3, 0, 0), (0, 3, 0), (0, 0, 3)Data from Prob. 3σ = e-x-y-z, T: 0 ≤ x ≤ 1 - y, 0 ≤ y ≤ 1, 0 ≤ z ≤ 2
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 normal vector N = ru × rv of the surface. Show the details of your work.r(u, v) = [α cos v,
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 = [xy, x2y2], C from (2, 0) straight to (0, 2)
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [ey, -ez, ex], S: x2 + y2 = 25, x ≥ 0, y ≥ 0, 0 ≤ z ≥ 2
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. (6, 1) e4(2x dx + 4x2 dy) (4,0)
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F as in Prob. 1, z = xy (0 ≤ x ≤ 1, 0 ≤ y ≤ 4). Compare with Prob. 1.Data from Prob. 1F = [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 = [x cosh 2y, 2x2 sinh 2y], R: x2 ≤ y ≤ x
Describe the region of integration and evaluate. 2 ,2x (r + y)² dy dx
Verify Theorem 1 for f = x2 - y2 and the surface of the cylinder x2 + y2 = 4, 0 ≤ z ≤ h.
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 normal vector N = ru × rv of the surface. Show the details of your work.xy-plane in polar
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [ey, ex, 1], S: x + y + z = 1, x ≥ 0, y ≥ 0, z ≥ 0
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 = [y2, -x2], C: y = 4x2 from (0, 0) to (1, 4)
Does the situation in Example 4 of the text change if you take the domain 0 < √x2 + y2 < 3/2?
Evaluate the surface integral ∫s∫ (curl F) • n dA directly for the given F and S.F = [-13 sin y, 3 sinh z, x], S the rectangle with vertices (0, 0, 2), (4, π/2, 2), (0, π/2, 2)
Evaluate ∫C F(r) • dr counterclockwise around the boundary C of the region R by Green’s theorem, whereF = [6y2, 2x - 2y4], R the square with vertices ±(2, 2), ±(2, -2)
Using b = u × p and (23), show that (when κ > 0)(23**) τ(s) = (u p p') = (r' r'' r''')/κ2.
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 velocity, speed, and tangential and normal acceleration.r(t) = [ct cos t, ct sin t, ct] (c ≠ 0)
Find the component of a in the direction of b. Make a sketch.What happens to the component of a in the direction of b if you change the length of b?
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 velocity, speed, and tangential and normal acceleration.r(t) = [2 cos t + cos 2t, 2 sin t - sin
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. FindDvf at P: (1, 1, 2)
Polar Coordinates Ï = ˆšx2+ y2, θ = arctan (y/n) givewhere Ï' = dÏ/dθ. Derive this. Use it to find the total length of cardioid Ï = α(1 - cos θ). Sketch this curve. p² + p² d0, 12
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Find(curl w) • v at (4, 0, 2)
Forces acting on moving objects (cars, airplanes, ships, etc.) require the engineer to know corresponding tangential and normal accelerations. In Probs. 35–38 find them, along with the velocity and speed. Sketch the path.r(t) = [8t, 6t, 0]. Find v and a.
If airplanes A and B are moving southwest with speed |vA| = 550 mph, and northwest with speed |vB| = 450 mph, respectively, what is the relative velocity v = vB - vA of B with respect to A?
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.Find the angle between a light ray and its reflection in three orthogonal plane mirrors, known as corner reflector.
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Findcurl v, curl w
Find the volume if the vertices are (1, 3, 6), (3, 7, 12), (8, 8, 9), and (2, 2, 8).
Evaluate the integral. Does Cauchy€™s theorem apply? Show details. У Re z dz, C: -1 -1 1 х
Integrate by a suitable method. | (Iz| + z) dz clockwise around the unit circle.
Integrate by the first method or state why it does not apply and use the second method. Show the details. Re z dz, C the parabola y = 1+ (x – 1)2 from 1+ i to 3 + 3i
Evaluate the integral. Does Cauchy€™s theorem apply? Show details. Ln (1 – z) dz, C the boundary of the parallelogram with vertices ti, ±(1 + i).
How would you find a bound for the left side in Prob. 19?Data from Prob. 19 \f{
Find a parametric representation and sketch the path.4(x - 2)2 + 5(y + 1)2 = 20
Show that ∫C (z - z1)-1 (z - z2)-1dz = 0 for a simple closed path C enclosing z1 and z2, which are arbitrary.
(a) If f(z) is not a constant and is analytic for all (finite) z, and R and M are any positive real numbers (no matter how large), show that there exist values of z for which |z| > R and |f(z)| > M. Use Liouville’s theorem.(b) If f(z) is a polynomial of degree n > 0 and M is an arbitrary
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(4z - 3)
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 figure). 3і 92i -3 х -3і
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 are in the cube of Prob. 11 at t = 1 occupy the volume e.Data from Prob. 11Show that the flow with
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 (gv)
Showing 3300 - 3400
of 3937
First
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Step by Step Answers
Get 50% OFF
Claim Now
