SOLUTIONS TO HOMEWORK ASSIGNMENT #2
1. The position of a particle is given by r(t) = 3 cos t, 4 cos t, 5 sin t , where > 0.
(a) Find the velocity vector v(t).
(b) Find the acceleration vector a(t).
(c) Find the speed v(t).
(d) True of false: a(t) = 2 r(t)
SOLUTIONS TO HOMEWORK ASSIGNMENT # 4
1. Evaluate the following line integrals using Greens theorem:
ydxxdy, where C is the circle x2 +y 2 = a2 oriented in the clockwise direction.
(a)
C
(b)
(y + x)dx + (x + sin y)dy, where C is any simple closed smooth cu
SOLUTIONS TO HOMEWORK ASSIGNMENT #1
1. Find the following limits.
et 1 1 + t 1 t 1
(a) limt0
,
,
.
t
t
t+1
1
(b) limt0+ et , t ln t, arctan
.
t
cos t 1
1
, cos
.
(c) lim 0,
2
t0
t
t
Solution:
et 1 1 + t 1 t 1
,
,
(a) limt0
=< 1, 1/2, 1 > .
t
t
t+1
1
=< 1,
SOLUTIONS TO HOMEWORK ASSIGNMENT # 6
1. Find a parametric representation of the following surfaces:
x
a
(a) that part of the ellipsoid
positive constants.
y
b
2
+
2
+
z
c
2
= 1 with y 0, where a, b, c are
(b) that part of the elliptical paraboloid x + y 2
SOLUTIONS TO MIDTERM #2, MATH 317
1. (9 marks) Answer true or false to the following statements by putting either true or
false in the boxes. Give valid reasons for your claims.
(a) If F is a conservative vector eld then divF = 0.
(b) The vector eld F = x
SOLUTIONS TO HOMEWORK ASSIGNMENT # 5
1. Determine whether or not the following vector elds are conservative. If it is conservative nd all potential functions.
(a) F = 3z 2 i + cos yj + 2xzk.
(b) F = y cos xyi + x cos xyj sin zk
Solution:By computation
i
j
SOLUTIONS TO MIDTERM #1, MATH 317
1. (9 marks) Answer true or false to the following questions by putting either true or false
in the boxes. If the answer is true give a proof or valid reason, and if the answer is
false state why.
(a) If C is a smooth spa