' (1) Find the real and positive constants and such that the following velocity field V is conservative V(x, y, z) = [21x sin (112)] + [√√ ₂²e-1]; + [x² cos(T2)-2 ze¯Y] K V (II) Consider a force field F(x,7₁²)=√(x₁4₁2) where is the conservative form of from part (I). Find & such that. F- (III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the points on 2-0 as source, sink or neither.
' (1) Find the real and positive constants and such that the following velocity field V is conservative V(x, y, z) = [21x sin (112)] + [√√ ₂²e-1]; + [x² cos(T2)-2 ze¯Y] K V (II) Consider a force field F(x,7₁²)=√(x₁4₁2) where is the conservative form of from part (I). Find & such that. F- (III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the points on 2-0 as source, sink or neither.
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
Please show the steps for first 3 questions.
![(1) Find the real and positive constants and such that the following velocity field V is conservative
V(x, y, z) = [21x sin (772)]; +[√T ₂²e-Y]; + [x² cos(172)_2₂€¯Y] K
V.
(II) Consider a force field F(x,7,₁²)=√(x₁4₁²) where is the conservative form of from part (1). Find & such that. F-
(III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the
points on 2-0 as source, sink or neither.
No structure operates in perfect isolation. Structures always interact with their environment and these interactions entail energy losses
which need to be minimised (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modelling of
components in renewable technologies concerns measures of the energy expended in the flows (e.g., air) around them. The work done
is one such measure.
(IV) Find the work done by F in moving a particle along any closed path C.
(V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10.
Find the work done by F in moving a particle along path C2 from r(O) to r(2).
(VI) Consider two paths C3 and C4, described by the parametric equations
#]
[²²] = α²₁ e-t [cont].
sint
't [sint.
cost
+aze-t
For simplicity, set a₁ = 0 and α²₂=1
(24)
[+] = α₁ [cat].
-2 sin (2t)
to define the equations for C3 and cy.
(g) Find the equation for C4, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
+α2
+
[
USE MATLAB to plot (3 and (y from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
sin (24)
(h) Find the work done by the force field & in moving a particle along path C4, from r(0) tor (Given
Gie) - Ezcas (2) E-toin234 (12sinh (tt)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12fdac56-4b52-4434-8b00-5772085cd8ad%2F8baf473d-aa46-403c-9184-59ce51860f31%2Fcq1sr9f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1) Find the real and positive constants and such that the following velocity field V is conservative
V(x, y, z) = [21x sin (772)]; +[√T ₂²e-Y]; + [x² cos(172)_2₂€¯Y] K
V.
(II) Consider a force field F(x,7,₁²)=√(x₁4₁²) where is the conservative form of from part (1). Find & such that. F-
(III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the
points on 2-0 as source, sink or neither.
No structure operates in perfect isolation. Structures always interact with their environment and these interactions entail energy losses
which need to be minimised (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modelling of
components in renewable technologies concerns measures of the energy expended in the flows (e.g., air) around them. The work done
is one such measure.
(IV) Find the work done by F in moving a particle along any closed path C.
(V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10.
Find the work done by F in moving a particle along path C2 from r(O) to r(2).
(VI) Consider two paths C3 and C4, described by the parametric equations
#]
[²²] = α²₁ e-t [cont].
sint
't [sint.
cost
+aze-t
For simplicity, set a₁ = 0 and α²₂=1
(24)
[+] = α₁ [cat].
-2 sin (2t)
to define the equations for C3 and cy.
(g) Find the equation for C4, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
+α2
+
[
USE MATLAB to plot (3 and (y from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
sin (24)
(h) Find the work done by the force field & in moving a particle along path C4, from r(0) tor (Given
Gie) - Ezcas (2) E-toin234 (12sinh (tt)
![MAST20029 Engineering Mathematics Formulae Sheet
1. Change of Variable of Integration in 2D
JJ₁ f(x, y) dxdy = [[ f(x(u, v), y(u, v))|J(u, v)| dudv
R
2. Transformation to Polar Coordinates
x = r cos 0,
3. Change of Variable of Integration in 3D
[[[ f(x, y, z) dadydz
4. Transformation to Cylindrical Coordinates
x = r cos 0, y = r sin 0,
5. Transformation to Spherical Coordinates
7. Work Integrals
y = r sin 0,
LFG
8. Surface Integrals
x = r cos sin o, y = r sin sin o, z = r cos o,
=
= [[[_ F(u, v), w)|J(u, v, w) dudvdw
[[₁, g(x, y, z) ds =
S
J(r,0) =
6. Line Integrals
Jo
f(x, y, 2) ds = ["* f(x(t), y(t), z(t)) √x'(t)² + y(t)}² + 2'(t)² dt
F(x, y, z). dr =
J(r, 0, z) = r
= r
cb dx
- [² R² +F="/
F₁ F₂
dt
a
F. ÂdS =
J(r, 0, 0) = r² sin o
9. Flux Integrals For a surface with upward unit normal,
11. F
+ F3
dz
= [[ g(x, y, f (x, y)) √[f² + f² +1dxdy
dt
= SS₁₂ - ²
-F₁fx - F2fy + F3 dydx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12fdac56-4b52-4434-8b00-5772085cd8ad%2F8baf473d-aa46-403c-9184-59ce51860f31%2F9x515i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:MAST20029 Engineering Mathematics Formulae Sheet
1. Change of Variable of Integration in 2D
JJ₁ f(x, y) dxdy = [[ f(x(u, v), y(u, v))|J(u, v)| dudv
R
2. Transformation to Polar Coordinates
x = r cos 0,
3. Change of Variable of Integration in 3D
[[[ f(x, y, z) dadydz
4. Transformation to Cylindrical Coordinates
x = r cos 0, y = r sin 0,
5. Transformation to Spherical Coordinates
7. Work Integrals
y = r sin 0,
LFG
8. Surface Integrals
x = r cos sin o, y = r sin sin o, z = r cos o,
=
= [[[_ F(u, v), w)|J(u, v, w) dudvdw
[[₁, g(x, y, z) ds =
S
J(r,0) =
6. Line Integrals
Jo
f(x, y, 2) ds = ["* f(x(t), y(t), z(t)) √x'(t)² + y(t)}² + 2'(t)² dt
F(x, y, z). dr =
J(r, 0, z) = r
= r
cb dx
- [² R² +F="/
F₁ F₂
dt
a
F. ÂdS =
J(r, 0, 0) = r² sin o
9. Flux Integrals For a surface with upward unit normal,
11. F
+ F3
dz
= [[ g(x, y, f (x, y)) √[f² + f² +1dxdy
dt
= SS₁₂ - ²
-F₁fx - F2fy + F3 dydx
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