Math 3202
Answers
Assignment #3
1. Given position vector r(t) = (et , et sin t, et cos t) , nd tangential and normal components of the
acceleration vector at (1, 0, 1).
Answer: From Homework 2, Problem 6b we know t = 0 as well as vectors T (0) and N (0).
Math 3202
Answers
Assignment #6
1. Evaluate the line integral along given curve
(a)
y ex ds, along the line segment jointing (1, 2) to (4, 7).
Solution Line segment has equations x = 1 + 3t, y = 2 + 5t, 0 t 1.
Then y ex ds = 01 (2 + 5t)e1+3t 34 dt = 34(16
Math 3202
Due Thur March 24
Assignment #7
1. Evaluate the line integral along given curve by two methods: (a) directly (b) using Greens Therem
(a)
(b)
C
xy 2 dx + x3 dy , where C is a rectangle with vertices (0, 0) (2, 0) (2, 3) (0, 3).
C x dx + y dy , wh
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
FINAL EXAM
Mathematics 3202
Fall 2003
n
Marks
[25]
1. The velocity of a charged particle moving in a magnetic eld and subject to the Lorentz
force is the vector function
v(t) = s
Math 3202
Solutions
Assignment #5
1. Evaluate the line integral of the vector eld
(a)
xy dx + (x y) dy, where C consists of line segments from (0,0) to (2,0) and from (2,0) to
(3,2);
Solution. First nd line integral I1 along segments from (0,0) to (2,0).
Math 3202
Answers
Assignment #6
This is just answers. Solutions will be distributed in class.
1. Evaluate the line integral by two methods: directly and using Greens Th
(a)
y dx x dy, where C is a circle of radius R.
Answer: 2R2 .
(b)
xy dx + x2 y 3 dy, w
Math 3202
Answers
Assignment #2
1. [3pt] Show that if a particle moves with a constant speed, then the velocity and acceleration vectors
are orthogonal.
2
2
2
Solution. v 2 = v v = v1 + v2 + v3
2
2
2
Given v1 + v2 + v3 =const, dierentiate both sides to ge
Math 3202
Solutions
Assignment #4, total=27 pt
1. Find the mass and the center of mass of lamina that occupies region D and has density (x, y).
Solution we use the formulas
m=
(x, y) dA,
x=
D
1
m
x(x, y) dA,
y=
D
1
m
y(x, y) dA,
D
(a) D = cfw_(x, y)|0 x a
Math 3202
Answers
Assignment #3
1. The position function of the spaceship is
r(t) = (cos t, sin t, tan t)
and the coordinates of the space station are R = ( 2, 2 2, 7).
a) At what moment of time t should the captain turn o the engines in order to coast in
Math 3202
Answers
Assignment #1
1. For each vector-function r(t) = (x(t), y(t)T , t [a, b] given below (a-d)
1. sketch x(t) and y(t)
2. sketch the curve in xy-plane, showing all details such as slope, asymptotes, vertices etc. Name
the curve.
3. nd the ve
Area between curves
Vertical slices.
The area between the curves y = f(x) and y = g(x) over a
b is obtained as follows.
x
b
A=
(f(x) g(x) dx
a
Observe that the rectangular slice has length given by f(x) g(x) and width dx.
y
y = f(x)
y = g(x)
a
b
x
dx
Hori
Coordinate Systems
Polar coordinates
x = r cos ,
y = r sin ,
and dA = r dr d
y
(x, y)
y
dA
rd
r
dr
d
r
x
r + dr
x
Cylindrical coordinates
x = r cos ,
y = r sin ,
z = z,
and dV = r dr d dz
z
z
rd
(x, y, z)
dr
dz
dV = r dr d dz
z
r
y
y
x
x
d
Spherical coord
Math 3202
Due Tue March 15
Assignment #6
1. Evaluate the line integral along given curve
(a)
y ex ds, along the line segment jointing (1, 2) to (4, 7).
(b)
(xy + ln x) dy , along the arc of the parabola y = x2 from (1, 1) to (3, 9).
(c)
x2 z ds, along the
Math 3202
Answers
Assignment #7
1. Evaluate the line integral along given curve by two methods: (a) directly (b) using Greens Therem
(a)
xy 2 dx + x3 dy , where C is a rectangle with vertices (0, 0) (2, 0) (2, 3) (0, 3).
Solution
(a) Line integral along e
Math 3202
Due Tue Feb 8
Assignment #3
1. Given position vector r(t) = (et , et sin t, et cos t) , nd tangential and normal components of the
acceleration vector. at (1, 0, 1).
2. The position function of the spaceship is
r(t) = (1 + t, 8 + t2 , 28 + t3 )
Math 3202
Due Tue March 1
Assignment #4
1. Find max and min values of the function f (x, y ) along the given curve. Plot the curve and the level
curves of the function in the vicinity of the points where the function achieves max or min values.
(a) f (x,
Math 3202
Answers
Assignment #4
1. Find max and min values of the function f (x, y ) along the given curve. Plot the curve and the level
curves of the function in the vicinity of the points where the function achieves max or min values.
(a) f (x, y ) = 4x
Math 3202
Answers
Assignment #2
1. Let trajectory be given by vector function r(t) = (t cos(2t), t sin(2t), 0). Find velocity vector v (t) =
r , angular momentum vector L(t) = r v , acceleration vector a(t) = r and torque vector
(t) = r a. Is there a mom
Math 3202
Due Tue Jan 25
Assignment #2
1. Let trajectory be given by vector function r(t) = (t cos(2t), t sin(2t), 0). Find velocity vector v (t) =
r , angular momentum vector L(t) = r v , acceleration vector a(t) = r and torque vector
(t) = r a. Is ther
Q
cfw_ x lyeal i x g
i g mWpy j~jcfw_g v ~ewcfw_ x g rxx Yzy|Wiywy j jxjkg4 uI
vru ~m7jxlzyjhgY|i|x3ujvg er~jcfw_ull lc jSv~"ywtwvyyr" g V v mV c
x m g i cfw_6 x g m x
v |ilyuerr~gybEH he rAewWGyredr ryy7ywWmV tsqn
ly GYal ~xerjeWrR [email protected]
cfw_ g x
Math 3202
Due Tue Jan 18
Assignment #1
1. Given parametric equations nd equation in the form F (x, y ) = 0 and sketch the curve. Do you
know its name?
(a) x = sec t, y = tan t, /2 < t < /2
(b) x = et , y = 1 et , ln 2 t ln 2
(c) x = 2 cos t, y = 3 sin t,
Math 3202
Answers
Assignment #1
1. Given parametric equations nd equation in the form F (x, y ) = 0 and sketch the curve. Do you
know its name?
(a) x = sec t, y = tan t, /2 < t < /2
Answer: 1 y 2 x2 = 0, x > 0, the right branch of hyperbola with vertex at
Math 3202
Due Tue March 8
Assignment #5
1. Find volume of the solid that lies within both the cylinder x2 + y 2 = 1 and the sphere x2 + y 2 + z 2 = 4.
2. Find the mass of a ball of radius a if the density at any point is proportional to its distance from
Math 3202
Solutions
Assignment #5
1. Find volume of the solid that lies within both the cylinder x2 + y 2 = 1 and the sphere x2 + y 2 + z 2 = 4.
Solution: In cylindrical coordinates the volume is bounded by cylinder r = 1 and sphere r2 + z 2 = 4.
4 r2
Thu
Math 3202
Solutions
Assignment #8(extra credit)
1. Is the following vector eld irrotational or incompressible at point (0, 1, 2) ?
F=
y
z
x
,2
,2
x2 + y 2 + z 2 x + y 2 + z 2 x + y 2 + z 2
Solution:
1. From Assignment 7 we know that curl F = 0 = (0, 0, 0)
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Mathematics 3202
Test 1
Answers
Instructor Margo Kondratieva
Student
Student number
Marks
1. A curve is given by a parametric equation. Name it out of the following list : Ellips
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Mathematics 3202
Test 2
Answers
Instructor Margo Kondratieva
Student
Student number
Marks
[10]
1. Let D be a region in the rst quadrant bounded by the curves x = 0, x = y and x2