Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2).
Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Change of Variable of Integration in 2D
[ f(x,y) drdy = f(z(u, v),y(u, 0)). («, ») ducho
2. Transformation to Polar Coordinates
The useful formulas
3. Change of Variable of Integration in 3D
[, (2, 2) dadydz = [[F(w, x, w)|J(u, v, w)| dududw
4. Transformation to Cylindrical Coordinates
z=rcos, y=rsin, Jr.)=r
6. Line Integrals
5. Transformation to Spherical Coordinates
x=rcos 0, y = rsin, ===, J(r,0,2)=r
x=rcos@sind, y=rsin@sind, 2=rcoso, Jr.,0,0)=²sin
7. Work Integrals
[1(x, y, z) ds = [ f(x(t), y(t),= (t)) √√x²(1)² + y′(t)² + 2′(t)²³ dt
8. Surface Integrals
[F(x, y, z) - dx = [° R² + R$/ + d
dr
dt
[ 92,9,2) ds = [[ 9(2.9.1(2.9)) √ 12 + 12 + 1 dady
9. Flux Integrals For a surface with upward unit normal,
J.P.
11. Stokes' Theorem
= [[₁-Fife - Faly + Pa dyda
F-nds=
10. Gauss' (Divergence) Theorem
JIL V. FdV =
dv = [[₁, F
[[
F. ÂdS
(V x F). ÂdS=
s = [ F..
F.dr
![Consider the real and positive constants n and y such that the following velocity field V is
conservative:
V(x, y, z) = [2nx sin(πz)] i+[√y z²e¯ ]] + [x² cos(z) - 2ze->] k
Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V
Question:
(d) Find the work done by F in moving a particle along any closed path C.
(e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving
through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving
a particle along path C₂ from r(0) to r(2).
We have already proved that:
n = 1 and y = 1
1
(2)
can let the velocity field V is conservative.
F = Vøp where p(x,y,z) = sin(xz) — 7² e-y
div(F) = -2ey < 0
therefore the points on the plane z = 0 are all sink.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fe47844-ccd9-47f2-b4da-7acecef93fac%2Fbc8fc367-9a08-4873-a9e2-3428dce9ecfa%2Ffqsu4cp_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the real and positive constants n and y such that the following velocity field V is
conservative:
V(x, y, z) = [2nx sin(πz)] i+[√y z²e¯ ]] + [x² cos(z) - 2ze->] k
Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V
Question:
(d) Find the work done by F in moving a particle along any closed path C.
(e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving
through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving
a particle along path C₂ from r(0) to r(2).
We have already proved that:
n = 1 and y = 1
1
(2)
can let the velocity field V is conservative.
F = Vøp where p(x,y,z) = sin(xz) — 7² e-y
div(F) = -2ey < 0
therefore the points on the plane z = 0 are all sink.
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