potential funeric (a) Explain why feF. dr has the same value for all three curves. (b) What is this common value? 1 heoret (0. L) and odi no 9010 (0) x adient is ations D 165 A 1457 YA 3+ 0 has such a function to det 1 have // 11 ++ 2 3 X summ libo12-18 (a) Find a function f such that F = Vf and (b) use or field. part (a) to evaluate F dr along the given curve C. .
potential funeric (a) Explain why feF. dr has the same value for all three curves. (b) What is this common value? 1 heoret (0. L) and odi no 9010 (0) x adient is ations D 165 A 1457 YA 3+ 0 has such a function to det 1 have // 11 ++ 2 3 X summ libo12-18 (a) Find a function f such that F = Vf and (b) use or field. part (a) to evaluate F dr along the given curve C. .
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
16.3
13 please
![1094
CHAPTER 16 Vector Calculus
0=()1 mintdo ow (LI) dow
002 02
vd 0/1 =
0
16.3 EXERCISES
1. The figure shows a curve C and a contour map of a function f
whose gradient is continuous. Find f Vf. dr.
XT
veleng
noney 2002
9. F(x, y) = (y² cos x + cos y)i + (2y sin x - x sin y) j
ne savent with bloft obrol auounimos & of 1stqedo eid 10. F(x, y) = (lny + y/x)i + (lnx + x/y) j
s
YA
brs
ni sdi 21 A
(stboronwishant (1)
X
0
1
2
C
y
10
0
1
3
2. A table of values of a function f with continuous gradient is
given. Find Jc Vf. dr, where C has parametric equations
x = 1² + 1
y = t³ + t
8
3+27(0
Comparing this equation with Equation 16, we see that
P(A) + K(A) = P(B) + K(B)
reason the vector field is called conservative.
which says that if an object moves from one point A to another point B under the influ-
ence of a conservative force field, then the sum of its potential energy and its kinetic
energy remains constant. This is called the Law of Conservation of Energy and it is the
on instemos A.A - (Onborant bo
I
15
or pelenk 50
40
30
20 Wib
1
6
5
WOND
60
2
4
7
29
silnopoA.D 19.
11.
Niog s te ((1))
3. F(x, y) = (xy + y²)i + (x² + 2xy) j
(A)A
4. F(x, y) = (y² - 2x) i + 2xy j
5. F(x, y) = y²e³ i + (1 + xy)e² j
6. F(x, y) = ye* i + (e* + e³) j
7.
F(x, y) = (ye* + sin y) i + (e* + x cos y) j
8. F(x, y) = (2xy + y 2)i + (x² - 2xy-³) j,
21 or 21
The figure shows the vector field F(x, y) = (2xy, x²) and
three curves that start at (1, 2) and end at (3, 2).
potential funch ((a) Explain why JcF. dr has the same value for all three
17
curves.
do do do ant ve anah, how si
dil(71) 26/10]
0 ≤ t ≤ 1 Men(htag
X
1-\V tadi balinov
ther
IFTO
be
y>0
liens ar il
(b) What is this common value?
y
3.
0
on sulder
70
+
1
en we have
2
sa bado 12. F(x, y) = (3 + 2xy²) i + 2x²y j,
مد
7
3706/103200
Jedi
of DYERVISA90 LX418d) ou 14. F(x, y) = (1 + xy)exy i + x²e* j,
//
12-18 (a) Find a function f such that F = Vf and (b) use
●
3-10 Determine whether or not F is a conservative vector field.part (a) to evaluate f F dr along the given curve C.
If it is, find a function f such that F = Vf.
(8)X = V
L
13. F(x, y) = x²y³ i + x³y²j,
ols blail cool erh yd snob show C: r(t) = (t³ - 2t, t³ + 2t), 0≤t≤1
bns SA
SATB ygrone
3
T
C is the arc of the hyperbola y = 1/x from (1, 1) to (4,1)
X
stalinatoq sut C: r(t) = cos ti + 2 sin tj, 0≤ t ≤ π/2
aved aw
1.-
15. F(x, y, z) = yzi+xzj + (xy + 2z) k,
C is the line segment from (1, 0, -2) to (4, 6, 3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1be8149a-7f70-4e17-bee2-140ac0078ae6%2F00ccddcc-d869-4e4f-bed8-b507e2ef044e%2F9554nr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1094
CHAPTER 16 Vector Calculus
0=()1 mintdo ow (LI) dow
002 02
vd 0/1 =
0
16.3 EXERCISES
1. The figure shows a curve C and a contour map of a function f
whose gradient is continuous. Find f Vf. dr.
XT
veleng
noney 2002
9. F(x, y) = (y² cos x + cos y)i + (2y sin x - x sin y) j
ne savent with bloft obrol auounimos & of 1stqedo eid 10. F(x, y) = (lny + y/x)i + (lnx + x/y) j
s
YA
brs
ni sdi 21 A
(stboronwishant (1)
X
0
1
2
C
y
10
0
1
3
2. A table of values of a function f with continuous gradient is
given. Find Jc Vf. dr, where C has parametric equations
x = 1² + 1
y = t³ + t
8
3+27(0
Comparing this equation with Equation 16, we see that
P(A) + K(A) = P(B) + K(B)
reason the vector field is called conservative.
which says that if an object moves from one point A to another point B under the influ-
ence of a conservative force field, then the sum of its potential energy and its kinetic
energy remains constant. This is called the Law of Conservation of Energy and it is the
on instemos A.A - (Onborant bo
I
15
or pelenk 50
40
30
20 Wib
1
6
5
WOND
60
2
4
7
29
silnopoA.D 19.
11.
Niog s te ((1))
3. F(x, y) = (xy + y²)i + (x² + 2xy) j
(A)A
4. F(x, y) = (y² - 2x) i + 2xy j
5. F(x, y) = y²e³ i + (1 + xy)e² j
6. F(x, y) = ye* i + (e* + e³) j
7.
F(x, y) = (ye* + sin y) i + (e* + x cos y) j
8. F(x, y) = (2xy + y 2)i + (x² - 2xy-³) j,
21 or 21
The figure shows the vector field F(x, y) = (2xy, x²) and
three curves that start at (1, 2) and end at (3, 2).
potential funch ((a) Explain why JcF. dr has the same value for all three
17
curves.
do do do ant ve anah, how si
dil(71) 26/10]
0 ≤ t ≤ 1 Men(htag
X
1-\V tadi balinov
ther
IFTO
be
y>0
liens ar il
(b) What is this common value?
y
3.
0
on sulder
70
+
1
en we have
2
sa bado 12. F(x, y) = (3 + 2xy²) i + 2x²y j,
مد
7
3706/103200
Jedi
of DYERVISA90 LX418d) ou 14. F(x, y) = (1 + xy)exy i + x²e* j,
//
12-18 (a) Find a function f such that F = Vf and (b) use
●
3-10 Determine whether or not F is a conservative vector field.part (a) to evaluate f F dr along the given curve C.
If it is, find a function f such that F = Vf.
(8)X = V
L
13. F(x, y) = x²y³ i + x³y²j,
ols blail cool erh yd snob show C: r(t) = (t³ - 2t, t³ + 2t), 0≤t≤1
bns SA
SATB ygrone
3
T
C is the arc of the hyperbola y = 1/x from (1, 1) to (4,1)
X
stalinatoq sut C: r(t) = cos ti + 2 sin tj, 0≤ t ≤ π/2
aved aw
1.-
15. F(x, y, z) = yzi+xzj + (xy + 2z) k,
C is the line segment from (1, 0, -2) to (4, 6, 3)
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