All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Ask a Question

Search
Sign In
Register

study help
mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Tangent Planes T(P) will be less important in our work, but you should know how to represent them.(a) If S: r(u, v), then T(P): (r* - r ru rv) = 0 (a scalar triple product) or
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Evaluate them with F or f and C as follows.F = [xz, yz, x2y2], C: r = [t, t, et], 0 ≤ t ≤ 5. Sketch C.
Construct three simple examples in each of which two equations (6') are satisfied, but the third is not.
Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [0
Same task as in Prob. 19 when w = x2 + y2 and C the boundary curve of the triangle with vertices (0, 0), (1, 0), (0, 1).Data from Prob. 19Show that w = ex sin y satisfies Laplace’s equation
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [xy, yz, zx], S the surface of the cone x2 + y2 ≤ 4z2, 0 ≤ z ≤ 2
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Elliptic cone z = √x2 + 4y2
Find Ix, Iy, I0of a mass of density f(x,y) = 1 in the region R in the figures, which the engineer is likely to need, along with other profiles listed in engineering handbooks.R as in Prob. 12.Data
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Evaluate them with F or f and C as follows.F = [ y1/3, x1/3, 0], C the hypocycloid r = [cos3 t, sin3 t, 0], 0 ≤ t ≤ π/4
Make a paper cross (Fig. 251) into a €œdouble ring€ by joining opposite arms along their outer edges (without twist), one ring below the plane of the cross and the other above.
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).(cos xy)(yz dx + xz dy) - 2 sin xy dz
Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [-y, 2z,
Show that for a solution w(x, y) of Laplace€™s equation ˆ‡2w = 0 in a region R with boundary curve C and outer unit normal vector n, LSE)-e)-« dx dy (12) дw -ds. Әn C. ||
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [cosh x, z, y], S as in Prob. 15Data from Prob. 15F = [2x2, 1/2y2, sin πz], S the
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Ellipsoid x2 + y2 + 19z2 = 1
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Find the center of gravity (xÌ…, yÌ…) of a mass of density f(x, y) = 1 in the given region R. х
Evaluate these integrals for the following data. Indicate the kind of surface. Show the details.G = arctan (y/x), S: z = x2 + y2, 1 ≤ z ≤ 9, x ≥ 0, y ≥ 0
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).ey dx + (xey - ez) dy - yez dz
Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [ey, 0,
Using (9), find the value of ∫C ∂w/∂n ds taken counterclockwise over the boundary C of the region R.W = x2 + y2, C: x2 + y2 = 4. Confirm the answer by direct integration.
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = as in Prob. 13, S the surface of x2 + y2 ≤ 9, 0 ≤ z ≤ 2Data from Prob. 13F =
Find a normal vector. The answer gives one representation; there are many. Sketch the surface and parameter curves.Plane 4x + 3y + 2z = 12
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Find the center of gravity (xÌ…, yÌ…) of a mass of density f(x, y) = 1 in the given region R.
Using (9), find a bound for the absolute value of the work W done by the force F = [x2, y] in the displacement from (0, 0) straight to (3, 4). Integrate exactly and compare.
Evaluate these integrals for the following data. Indicate the kind of surface. Show the details.G = αx + by + cz, S: x2 + y2 + z2 = 1, y = 0, z = 0
Check, and if independent, integrate from (0, 0, 0) to (α, b, c).(sinh xy) (z dx - x dz)
Calculate this line integral by Stokes’s theorem for the given F and C. Assume the Cartesian coordinates to be right-handed and the z-component of the surface normal to be nonnegative.F = [z3, x3,
Using (9), find the value of ∫C ∂w/∂n ds taken counterclockwise over the boundary C of the region R.w = x2y + xy2, R: x2 + y2 ≤ 1, x ≥ 0, y ≥ 0
The importance of the divergence theorem in potential theory is obvious from (7)€“(9) and Theorems 1€“3. To emphasize it further, consider functions f and g that are harmonic in
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.F = [x3 - y3, y3 - z3, z3 - x3], S the surface of x2 + y2 + z2 ≤ 25, z ≥ 0
Find the points in Probs. 1–8 at which (4) N ≠ 0 does not hold. Indicate whether this results from the shape of the surface or from the choice of the representation.
Evaluate ∫C F (r) • dr for given F and C by the method that seems most suitable. Remember that if F is a force, the integral gives the work done in the displacement along C. Show
Find the center of gravity (xÌ…, yÌ…) of a mass of density f(x, y) = 1 in the given region R. h R
Consider the integral ∫C F(r) • dr, where F = [xy, -y2].(a) Find the value of the integral when r = [cos t, sin t], 0 ≤ t ≤ π/2. Show that the value remains the same if you set t =
Evaluate these integrals for the following data. Indicate the kind of surface. Show the details.G = cos x + sin x, S the portion of x + y + z = 1 in the first octant
Integrate x2y dx + 2xy2 dy over various circles through the points (0, 0) and (1, 1). Find experimentally the smallest value of the integral and the approximate location of the center of the circle.
Let R and C be as in Green€™s theorem, r' a unit tangent vector, and n the outer unit normal vector of C (Fig. 240 in Example 4). Show that (1) may be writtenorwhere k is a unit vector
Use the divergence theorem, assuming that the assumptions on T and S are satisfied.Show that a region T with boundary surface S has the volumewhere r is the distance of a variable point P: (x, y, z)
Evaluate the surface integral ∫∫S F • n dA by the divergence theorem. Show the details.Solve Prob. 9 by direct integration.Data form Prob. 9F = [x2, 0, z2], S the surface of the box
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, -z, 2y] from (0, 0, 0) straight
Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.F = [y2, x2, z4], S: z = 4 √x2 + y2, 0 ≤ z ≤ 8, y ≥ 0
(a) Show that I = ˆ«C (x2y dx + 2xy2dy) is path dependent in the xy-plane.(b) Integrate from (0, 0) along the straight-line segment to (1, b), 0 ‰¤ b ‰¤ 1, and
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 = Ï€Î±2 h/3 verify for the volume of a
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
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,
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
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
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
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
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 =
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 = [xy, x2y2], C from (2, 0) straight to
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
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],
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
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
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 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
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, Î¸ = arctan (y/n) givewhere Ï' = dÏ/dÎ¸. Derive this. Use it to find the total length of cardioid
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
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
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)|
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(4z - 3)

Showing 3200 - 3300 of 3883

First
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Company Info

Security
Copyrights
Privacy Policy
Terms & Condition
SolutionInn Fee
Scholarship

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Copyright © 2024 SolutionInn All Rights Reserved.

Get the SolutionInn - Study Help App