1
THE UNIVERSITY OF BRITISH COLUMBIA
Sessional Examinations – April 2010
MATHEMATICS 317
TIME: 2.5 hours
NO AIDS ARE PERMITTED. Each question is of equal value and is worth 10 points.
Note that the maximum number of points is 70. A score of N/70 will be treated as
N/55. Also note that this exam has two pages.
1. Consider the function
.),( xyyxf
=
(a) Explicitly determine the field lines (flow lines) of .),( fyx
∇
=
F
(b) Sketch the field lines of F and the level curves of
f
in the same diagram.
2. Suppose, in terms of the time parameter
t
, a particle moves along the path
.1,)sin(cos)cos(sin)(
2
∞<≤+++−= ttttttttt
kjir
(a) Find the speed of the particle at time
t
.
(b) Find the tangential component of acceleration at time
t
.
(c) Find the normal component of acceleration at time
t
.
(d) Find the curvature of the path at time
t
.
3. Let
k
j
i
r
)()()()(
tztytxt
+
+
=
be the position of a particle at time
t
. Suppose the
motion of the particle satisfies the differential equation r
r)(
2
2
rf
dt
d=
where
.||
r
=
r
(a) Suppose
f
(
r
) is an arbitrary function of
r
. Prove or disprove each of the
following statements.
(i) The motion of the particle is planar.
(ii) The path of the particle sweeps out equal areas in equal times.
(b) Find all forms of
f
(
r
) for which the motion of the particle always lies on a
straight line.
(c) Give a specific form of
f
(
r
) for which the motion of the particle could lie on
an ellipse.
4. Let
3
/),,(
rzyx
rF
=
where
k
j
i
r
zyx
+
+
=
and .||
r
=
r
(a) Find
.F
⋅
∇
(b) Find the flux of F outwards through the spherical surface .
2222
azyx
=++
(c) Do the results of (a) and (b) contradict the divergence theorem? Explain your
answer.
(d) Let
E
be the solid region bounded by the surfaces 1,01
222
==+−−
zyxz
and .1
−
=
z Let
σ
be the bounding surface of E. Determine the flux of F
outwards through
σ
.
(e) Let R be the solid region bounded by the surfaces
1,034
222
==−+−− zyyxz and .1
−
=
z Let
Σ
be the bounding surface
of R. Determine the flux of F outwards through
.
Σ
