h 13. F(x, y, z)=-yi+xj-2k, be S is the cone z² = x² + y²,0 ≤ z ≤ 4, oriented downward
h 13. F(x, y, z)=-yi+xj-2k, be S is the cone z² = x² + y²,0 ≤ z ≤ 4, oriented downward
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
16.8
13&14 please
![H
16.8 EXERCISES
ved bas
232122571
1. A hemisphere H and a portion P of a paraboloid are shown.
Suppose F is a vector field on R³ whose components have
continuous partial derivatives. Explain why (8)
SS
Val
curl F. ds = curl F. ds
the pluvely ore 11. (a) Use Stokes' Theorem to evaluate jc F • dr, where
and this theorem to veci
Tu
F(x, y, z) = x²zi + xy²j + z²k
ff
H
ZA
4
P
(alvis broci so nwond o
en szarlo tod le
12 U
noi
bong bne nisig
br
ola bili to
2 bnn man
you 5x4
x4
m-1
2y
2
2
ZA
2 4 ЯОЗНТ OWT GIVA I and
poputing216
odtown
2-6 Use Stokes' Theorem to evaluate ff, curl F. ds.
tion.
Mit
CIL
2 y
2. F(x, y, z) = x² sin zi + y² j + xy k,
S is the part of the paraboloid z = 1 = x² - y² that lies
above the xy-plane, oriented upward
3. F(x, y, z) = zei + x cos y j + xz sin y k,
S is the hemisphere x² + y² + z² = 16, y ≥ 0, oriented in
the direction of the positive y-axis
.
12. (a) Use Stokes' Theorem to evaluate f F dr, where
F(x, y, z) = x²yi + x³j + xy k and C is the curve of
intersection of the hyperbolic paraboloid z = y² - x²
and the cylinder x² + y² = 1, oriented counterclock-
Insie wise as viewed from above.
ISLeigalup
(b) Graph both the hyperbolic paraboloid and the cylinder
Senidorul vo
Holood sd with domains chosen so that you can see the curve C
a and the surface that you used in part (a).
lovisa bo] lo piqe
(c) Find parametric equations for C and use them to
graph C.
4. F(x, y, z) = tan¯¹(x²yz²) i + x²yj + x²z² k, o visiboz, sudgisbelid
S is the cone x = √y² + z², 0≤x≤ 2, oriented in the
direction of the positive x-axis
6. F(x, y, z) = e¹ i + e™²j + x²zk,
S is the half of the ellipsoid 4x² + y² + 4z² = 4 that lies to
the right of the
xz-plane, oriented in the direction of the
exz-p
ref
1940 bair
positive y-axis
7. F(x, y, z)=(x + y²)i + (y + z²)j + (z + x²) k,
C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)
8. F(x, y, z)= i + (x + yz)j + (xy-√√z)k,
C is the boundary of the part of the plane 3x + 2y + z = 1
in the first octant
SECTION 16.8 Stokes' Theorem
GEROUS DI
DIA
nuomoto
7-10 Use Stokes' Theorem to evaluate F. dr. In each case C
is oriented counterclockwise as viewed from above.
9. F(x, y, z)= xy i+yzj + zx k, C is the boundary of the
part of the paraboloid z = 1 - x² - y² in the first octant
xb (2.00
Al
10. F(x, y, z) = 2yi+xz j + (x + y) k, C is the curve
of intersection of the plane z = y + 2 and the cylinder
x² + y² = 1
vid
and C is the curve of intersection of the planet
x + y + z = 1 and the cylinder x2 + y² = 9, oriented
counterclockwise as viewed from above.
13. F(x, y, z) = -yi+xj-2 k,
sitesinto consideS is the cone z² = x² + y²,0 ≤ z ≤ 4, oriented downward
gela
mont bos stins all ne X-omule si all. W 19
Line
14. F(x, y, z) -2yzi+yj + 3x k,
M
=
5. F(x, y, z) = xyz i + xy j + x²yz k, IDX sinuto puroS is the part of the paraboloid z = 5 - x² - y² that lies
S consists of the top and the four sides (but not the bottom)
Mat
above the plane z 1, oriented upward
=
of the cube with vertices (±1, ±1, ±1), oriented outward det
doM AM 15.
(b) Graph both the plane and the cylinder with domains
chosen so that you can see the curve C and the surface
that you used in part (a).
(c) Find parametric equations for C and use them to
graph C.
AL 700
1139
13-15 Verify that Stokes' Theorem is true for the given vector
field F and surface S.
F(x, y, z) = yi + zj+xk,
S is the hemisphere x² + y² + z² = 1, y = 0, oriented in
the direction of the positive y-axis
16. Let C be a simple closed smooth curve that lies in the plane
x + y + z = 1. Show that the line integral
S z dx
-
2x dy + 3ydz
depends only on the area of the region enclosed by C and
not on the shape of C or its location in the plane.
Find the work done.
17. A particle moves along line segments from the origin to the
points (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the origin
under the influence of the force field
F(x, y, z) = z² i + 2xy j + 4y² k](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1be8149a-7f70-4e17-bee2-140ac0078ae6%2F5eeb6687-5dab-454c-a6cd-34a13a711070%2Fthmo4ri_processed.jpeg&w=3840&q=75)
Transcribed Image Text:H
16.8 EXERCISES
ved bas
232122571
1. A hemisphere H and a portion P of a paraboloid are shown.
Suppose F is a vector field on R³ whose components have
continuous partial derivatives. Explain why (8)
SS
Val
curl F. ds = curl F. ds
the pluvely ore 11. (a) Use Stokes' Theorem to evaluate jc F • dr, where
and this theorem to veci
Tu
F(x, y, z) = x²zi + xy²j + z²k
ff
H
ZA
4
P
(alvis broci so nwond o
en szarlo tod le
12 U
noi
bong bne nisig
br
ola bili to
2 bnn man
you 5x4
x4
m-1
2y
2
2
ZA
2 4 ЯОЗНТ OWT GIVA I and
poputing216
odtown
2-6 Use Stokes' Theorem to evaluate ff, curl F. ds.
tion.
Mit
CIL
2 y
2. F(x, y, z) = x² sin zi + y² j + xy k,
S is the part of the paraboloid z = 1 = x² - y² that lies
above the xy-plane, oriented upward
3. F(x, y, z) = zei + x cos y j + xz sin y k,
S is the hemisphere x² + y² + z² = 16, y ≥ 0, oriented in
the direction of the positive y-axis
.
12. (a) Use Stokes' Theorem to evaluate f F dr, where
F(x, y, z) = x²yi + x³j + xy k and C is the curve of
intersection of the hyperbolic paraboloid z = y² - x²
and the cylinder x² + y² = 1, oriented counterclock-
Insie wise as viewed from above.
ISLeigalup
(b) Graph both the hyperbolic paraboloid and the cylinder
Senidorul vo
Holood sd with domains chosen so that you can see the curve C
a and the surface that you used in part (a).
lovisa bo] lo piqe
(c) Find parametric equations for C and use them to
graph C.
4. F(x, y, z) = tan¯¹(x²yz²) i + x²yj + x²z² k, o visiboz, sudgisbelid
S is the cone x = √y² + z², 0≤x≤ 2, oriented in the
direction of the positive x-axis
6. F(x, y, z) = e¹ i + e™²j + x²zk,
S is the half of the ellipsoid 4x² + y² + 4z² = 4 that lies to
the right of the
xz-plane, oriented in the direction of the
exz-p
ref
1940 bair
positive y-axis
7. F(x, y, z)=(x + y²)i + (y + z²)j + (z + x²) k,
C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)
8. F(x, y, z)= i + (x + yz)j + (xy-√√z)k,
C is the boundary of the part of the plane 3x + 2y + z = 1
in the first octant
SECTION 16.8 Stokes' Theorem
GEROUS DI
DIA
nuomoto
7-10 Use Stokes' Theorem to evaluate F. dr. In each case C
is oriented counterclockwise as viewed from above.
9. F(x, y, z)= xy i+yzj + zx k, C is the boundary of the
part of the paraboloid z = 1 - x² - y² in the first octant
xb (2.00
Al
10. F(x, y, z) = 2yi+xz j + (x + y) k, C is the curve
of intersection of the plane z = y + 2 and the cylinder
x² + y² = 1
vid
and C is the curve of intersection of the planet
x + y + z = 1 and the cylinder x2 + y² = 9, oriented
counterclockwise as viewed from above.
13. F(x, y, z) = -yi+xj-2 k,
sitesinto consideS is the cone z² = x² + y²,0 ≤ z ≤ 4, oriented downward
gela
mont bos stins all ne X-omule si all. W 19
Line
14. F(x, y, z) -2yzi+yj + 3x k,
M
=
5. F(x, y, z) = xyz i + xy j + x²yz k, IDX sinuto puroS is the part of the paraboloid z = 5 - x² - y² that lies
S consists of the top and the four sides (but not the bottom)
Mat
above the plane z 1, oriented upward
=
of the cube with vertices (±1, ±1, ±1), oriented outward det
doM AM 15.
(b) Graph both the plane and the cylinder with domains
chosen so that you can see the curve C and the surface
that you used in part (a).
(c) Find parametric equations for C and use them to
graph C.
AL 700
1139
13-15 Verify that Stokes' Theorem is true for the given vector
field F and surface S.
F(x, y, z) = yi + zj+xk,
S is the hemisphere x² + y² + z² = 1, y = 0, oriented in
the direction of the positive y-axis
16. Let C be a simple closed smooth curve that lies in the plane
x + y + z = 1. Show that the line integral
S z dx
-
2x dy + 3ydz
depends only on the area of the region enclosed by C and
not on the shape of C or its location in the plane.
Find the work done.
17. A particle moves along line segments from the origin to the
points (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the origin
under the influence of the force field
F(x, y, z) = z² i + 2xy j + 4y² k
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