(a) Find the real and positive constants and y such that the following velocity field V is conservative V(x, y, z)= [2nx sin(xz)] 1+ [√y z²e¯ ]] + [x² cos(z) – 2ze">] k
(a) Find the real and positive constants and y such that the following velocity field V is conservative V(x, y, z)= [2nx sin(xz)] 1+ [√y z²e¯ ]] + [x² cos(z) – 2ze">] k
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V
from part (a). Find such that F = V.
(a) Find the real and positive constants and y such that the following velocity field V is
conservative
V(x, y, z) = [2nx sin(rz)] 1+ [√y z²ey] + [x² cos(rz) - 2ze->] k
(c) Can the divergence of F be zero on the plane z=0? Justify your answer using the
divergence of F on this plane. Classify the points on z=0 as source, sink or neither.
(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:
(f) Consider two paths C3 and C, described by the parametric equations in your NOTES
on pages 2.25 and 2.27, respectively (go to the "general solution" to the system of
ordinary differential equations that is given on each page).*
(g) Find the equation for C, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
For simplicity, set a₁ = 0 and a2 = 1 to define the equations for C₁ and C₂.
For example, on page 2.25, the path C, would be parametrised by
r(t) = [e-t sin(t)] i+ [e-* cos(t)] j+[0] k
THE UNIVERSITY OF
Use MATLAB to plot C3 and C₁ from r(0) to r() indicating the direction of
increasing t (you may manually add this direction in your plot).
The solutions of (a) (b) (c) (d) (e)are given as references, please solve for
(f) and (g),i will be very grateful!!
(a)
We have already proved that:
n = and y = 1
(b)
can let the velocity field V is conservative.
F = V¢ where p(x,y,z) = sin(xz) — 2² e-y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fe47844-ccd9-47f2-b4da-7acecef93fac%2Fd6dc881f-80dd-4541-84e6-e6e0bcd6896b%2F2b0xtjp_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V
from part (a). Find such that F = V.
(a) Find the real and positive constants and y such that the following velocity field V is
conservative
V(x, y, z) = [2nx sin(rz)] 1+ [√y z²ey] + [x² cos(rz) - 2ze->] k
(c) Can the divergence of F be zero on the plane z=0? Justify your answer using the
divergence of F on this plane. Classify the points on z=0 as source, sink or neither.
(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:
(f) Consider two paths C3 and C, described by the parametric equations in your NOTES
on pages 2.25 and 2.27, respectively (go to the "general solution" to the system of
ordinary differential equations that is given on each page).*
(g) Find the equation for C, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
For simplicity, set a₁ = 0 and a2 = 1 to define the equations for C₁ and C₂.
For example, on page 2.25, the path C, would be parametrised by
r(t) = [e-t sin(t)] i+ [e-* cos(t)] j+[0] k
THE UNIVERSITY OF
Use MATLAB to plot C3 and C₁ from r(0) to r() indicating the direction of
increasing t (you may manually add this direction in your plot).
The solutions of (a) (b) (c) (d) (e)are given as references, please solve for
(f) and (g),i will be very grateful!!
(a)
We have already proved that:
n = and y = 1
(b)
can let the velocity field V is conservative.
F = V¢ where p(x,y,z) = sin(xz) — 2² e-y
![(c)
div(F) = -2e < 0
therefore the points on the plane z = 0 are all sink.
(d) the work done by the force field l' to move a particle along a closed path C' is zero
(e)If the work done by the force field to move a particle along path C is 10, the work done to
move the particle along path C is also 10, regardless of the difference in the
MAST20029 Engineering Maths: Systems of ODES 2.25
paths.
Case 5: A = a +bi (a‡0)
Exercise 8
Classify the critical point at the origin and sketch the phase
portrait for the system:
dx
dt
dy
dt
Solution
The general solution is
[:] -
are
=
-x+y
-x-y
cost
sin t
+ age"
-
Solution
The general solution is
[:] - - [
=
MAST20029 Engineering Maths: Systems of ODEs
dx
Case 6: A = ±8i
Exercise 9
Classify the critical point at the origin and sketch the phase
portrait for the system:
sint
cost
cos(21)
y
-4x
+02
2.27
sin(2t)
2 cos (21)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fe47844-ccd9-47f2-b4da-7acecef93fac%2Fd6dc881f-80dd-4541-84e6-e6e0bcd6896b%2F80wyqq8_processed.png&w=3840&q=75)
Transcribed Image Text:(c)
div(F) = -2e < 0
therefore the points on the plane z = 0 are all sink.
(d) the work done by the force field l' to move a particle along a closed path C' is zero
(e)If the work done by the force field to move a particle along path C is 10, the work done to
move the particle along path C is also 10, regardless of the difference in the
MAST20029 Engineering Maths: Systems of ODES 2.25
paths.
Case 5: A = a +bi (a‡0)
Exercise 8
Classify the critical point at the origin and sketch the phase
portrait for the system:
dx
dt
dy
dt
Solution
The general solution is
[:] -
are
=
-x+y
-x-y
cost
sin t
+ age"
-
Solution
The general solution is
[:] - - [
=
MAST20029 Engineering Maths: Systems of ODEs
dx
Case 6: A = ±8i
Exercise 9
Classify the critical point at the origin and sketch the phase
portrait for the system:
sint
cost
cos(21)
y
-4x
+02
2.27
sin(2t)
2 cos (21)
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