430
CHAPTER 45
45.42
Evaluate L = /^ y dx + z dy — x dz, where <£ is the line segment from (0,1, -1) to (1,2,1).
Parametric equations for <# are x = t, y — l + t, z = —l + 2t for Osr^l. Hence,
45.43
Find the work W done by a force F = (xy, yz, xz) acting on an object moving along the curve R(f) = (t. t
2 , t
3 )
for 0
Let IS be any curve in the *_y-plane. Show that j^ y dx + x dy depends only on the endpoints of
( 6.
45.45
Prove that any gradient field Vf in a region is conservative. More precisely, if ^ is any curve in the region from a
point P to a point Q, then ^ Vf-tds
=f(Q)-f(P).
also holds if any two points in the region can be connected by a continuous curve lying in the region.)
(Note: The converse
If « is given by (x(t), y(t), z(t)) for o
If < 2 + y
2 = 49, F(JC, y) = (x
2 , y
2 ), and n denotes the exterior unit normal vector, compute
j^ F • n ds.
which is the correct equation
We have div F=l + l + l=3 and, on *#, F = an. Hence, the theorem reads
Gauss' theorem, JJ F • n dS = JJJ div F dV, where ^ is a closed surface bounding a three-dimensional region
&i, is the analogue to Green's theorem in the plane. Verify Gauss' theorem for the surface ^: x
2 + y
2 + z
: = a
2
and the vector field F = (x, y, z).
45.49
45.48
45.47
State Green's theorem in the plane.
Verify Green's theorem (Problem 45.47) when F = (3x, 2y) and 2ft is the disk of radius 1 with center at the
origin.
Therefore,
Then,
circle x - cos f, y = sin /, 0 £ t s 2-rr.
div
% is the
If 12 is a closed plane curve bounding a region SI and n denotes the exterior unit normal vector along
< €. then
J"^ F • n ds = J J div F dA for any vector field F on 01.
*
since
Hence, we obtain J^ x
2 dy - >'
: dx =
where ^ is parametrized as ;c = 7cosr, y = 7sin/, Osr<2?r. Thus, we have
(by periodicity).
CHAPTER 45
45.42
Evaluate L = /^ y dx + z dy — x dz, where <£ is the line segment from (0,1, -1) to (1,2,1).
Parametric equations for <# are x = t, y — l + t, z = —l + 2t for Osr^l. Hence,
45.43
Find the work W done by a force F = (xy, yz, xz) acting on an object moving along the curve R(f) = (t. t
2 , t
3 )
for 0
Let IS be any curve in the *_y-plane. Show that j^ y dx + x dy depends only on the endpoints of
( 6.
45.45
Prove that any gradient field Vf in a region is conservative. More precisely, if ^ is any curve in the region from a
point P to a point Q, then ^ Vf-tds
=f(Q)-f(P).
also holds if any two points in the region can be connected by a continuous curve lying in the region.)
(Note: The converse
If « is given by (x(t), y(t), z(t)) for o
If < 2 + y
2 = 49, F(JC, y) = (x
2 , y
2 ), and n denotes the exterior unit normal vector, compute
j^ F • n ds.
which is the correct equation
We have div F=l + l + l=3 and, on *#, F = an. Hence, the theorem reads
Gauss' theorem, JJ F • n dS = JJJ div F dV, where ^ is a closed surface bounding a three-dimensional region
&i, is the analogue to Green's theorem in the plane. Verify Gauss' theorem for the surface ^: x
2 + y
2 + z
: = a
2
and the vector field F = (x, y, z).
45.49
45.48
45.47
State Green's theorem in the plane.
Verify Green's theorem (Problem 45.47) when F = (3x, 2y) and 2ft is the disk of radius 1 with center at the
origin.
Therefore,
Then,
circle x - cos f, y = sin /, 0 £ t s 2-rr.
div
% is the
If 12 is a closed plane curve bounding a region SI and n denotes the exterior unit normal vector along
< €. then
J"^ F • n ds = J J div F dA for any vector field F on 01.
*
since
Hence, we obtain J^ x
2 dy - >'
: dx =
where ^ is parametrized as ;c = 7cosr, y = 7sin/, Osr<2?r. Thus, we have
(by periodicity).