University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2013/2014
Assignment #3
This assignment is due
February 4 February 7, 2014.
A. Suggested reading:
at
the
start
of
your
tutorial
in
the
1.
Marsden & Tromba, Chapter 2
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #7
1. Put = F1 dx + F2 dy + F3 dz = y 2 z 3 dx + 2xyz 3 dy +
F2 F1
F1
= 2yz 3 =
,
=
3xy 2 z 2 dz. We note that
y
x z
F2
F3
F3
and
= 6xyz 2 =
, so
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #6
1. (a) Since is a closed path oriented in the counterclockwise direction and F
Z is defined
and C 1 on the enclosed region we can use Greens T
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #9
1. We parametrize the 4 edges as 1 (x) = (x, a2 ), a1 x b1 , n1 = (0, 1); 2 (y) =
(b1 , y), a2 y b2 , n2 = (1, 0); 3 (x) = (x, b2 ), a1 x b1 ,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #3
1. (t) = (4t, t2 , t3 ) so 0 (t) = (4, 2t, 3t2 ). After
leaving the path the particle follows the tangent line
(2) + s 0 (2), s 0. When t = 4,
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
Problem Set II Solution Sketch
1. Find the entire area bounded by the polar graph r 2 = a2 sin 3.
Solution:
Since each lobe is the same by symmetry,
A=3
Z
0
/3 Z a sin 3
r dr dA = 3
Z
/3
0
0
r=asin 3
Z /3
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #4
1. (a) Since kT (t)k = 1, we have kT (t)k2 = T (t) T (t) = 1. Differentiating w.r.t.
d
t gives
(T (t) T (t) = T (t) T (t) + T (t) T (t) = 2 T
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
Problem Set V Solution Sketch
1. Compute the line integral of = x dx + y 3 dy along the curve
(t) = e ln cos(t/4) , (sin t) arctan(t 1) ,
t
0 t 1.
Solution:
4
2
By inspection,
x dx + y 3dy = dg where
g(x,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #5
1. (a) = 6x2 y dx + 2x3 dy . Put F1 (x, y) = 6x2 y and F2 (x, y) = 2x3 . Since
F1
= 6x2
y
F2
= 6x2 , is closed. Since F1 and F2 are defined fo
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University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MAT B42H
2005/2006
Solutions #2
.
1. f (x) has period and corresponds to sin x on 0,
2
2
2
Z
2 2
4
=
a0 =
cos x
sin x dx =
0
2 0
Z
4
2 2
1
.
ak =
sin x cos(4kx) dx
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42F
Problem Set IX Solution Sketch
R
1. Use Gauss (Divergence) Theorem to calculate
F n dS where
F(x, y, z) = (3x2 , xy, z),
is the surface which bounds the set cfw_(x, y, z) | x + y + z 1, x 0, y 0, z 0,
and
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42F
1. (page 515, # 12c) Find
0 z (x2 + y 2 )1/2 .
Problem Set VIII Solution Sketch
R
S
xdS where S is the part of the cylinder x2 + y 2 = 2x with
Remark: My interpretation of the question is that the cylinder
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2013/2014
Assignment #5
The Term Test will take place on Monday, March 3, 5:00 7:00 pm .
This assignment is due at
February 25 February 28, 2014.
the
start
of
your
t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #11
2
, t sin t +
, 0 t 2. Now
1. We can parametrize by (t) = t2 cos t +
2
2
(t) = 2t cos t +
t2 sin t +
, 2t sin t +
+
4 2
2
2
2
t2 cos t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #8
1. We can start by drawing the centerline ` lengthwise along the strip.
A
C
B
A
cfw_
C
B
When A is attached to B 0 and B is attached to A0 , w
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42F
Problem Set VII Solution Sketch
1. Compute the surface area of the piece of the plane z = y +1 which lies over the interior
of the circle x2 + y 2 = 1.
Solution:
Let B be the interior of the unit circle an
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
1.)
Problem Set I Solution Sketch
a) In each of the following, sketch the curve in R2 whose polar coordinates satisfy
the given equation.
(i) r = cos
Solution:
r = cos
Equivalently, r 2 = r cos
Equivale
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #1
1. (a) f (x) = 2 x, x < .
Since f (x) is an odd Zfunction, ak = 0,
2
1
even
k = 0, 1, 2, . bk =
2x sin kx dx =
Z
4
1
4
parts
x sin kx dx =
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
Problem Set III Solution Sketch
1. Give a parameterization and a description of the following curves. In part (b) give a
description and/or sketch of the curve.
a) The intersection of the bowl z = 4x2 + 9y
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2009/2010
Solutions #2
x ,0 x < 1
0 ,1 x < 2
and extended from this with period 2
to the rest of R.Z
Since the period
1
2 2
is 2, ak =
f (x) cos(kx) dx =
2
0
Z 1
x s
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
Problem Set IV Solution Sketch
1. (page 272; #9, #10, #11) Sketch a few flow lines of the vector fields
a) F(x, y) = (y, x)
Solution:
b) F(x, y) = (x, y)
Solution:
c) F(x, y) = (x, x2 )
Solution:
1
2. Let
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42S
Problem Set VI Solution Sketch
1. Use Greens
R 2 Theorem to evaluate the following line integrals:
a) x y dx x dy where proceeds clockwise around the boundary of the region
cfw_(x, y) | 3 x 5, 1 y 3.
Solut
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42F
Problem Set XI Solution Sketch
1. Let = 3z dx+4x dy+2y dz and let be the piece of the surface z = 9x2 y 2 for z
0. Let be the circleR x2 + y 2 = 9 lying in the xy-plane, taken in the counterclockwise
dire
SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
MAT B42F
Problem Set X Solution Sketch
1. (page 604, #4) Let V be a vector field defined by
V(x, y, z) = (G(x, y, z), H(x, y, z), F (x, y, z)
where F , G, and H are functions into R, and let be the 2-form given by
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+,
-./ 10
2&4350-'!6 7/.8:9 ;1*)<0;1=>/?@>=<."
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University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MAT B42H
2007/2008
Solutions #7
1. Put = F1 dx + F2 dy + F3 dz = y 2z 3 dx + 2xyz 3 dy +
F2 F1
F1
= 2yz 3 =
,
=
3xy 2z 2 dz. We note that
y
x z
F2
F3
F3
3y 2 z 2 =
and
= 6x
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MAT B42H
2005/2006
Solutions #8
1. We can start by drawing the centerline ` lengthwise along the strip.
A
C
B
A
cfw_
C
B
When A is attached to B 0 and B is attached to A
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Assignment #7
This assignment is due
March 6 March 10, 2017.
at
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Chapter 7,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Assignment #5
The Term Test will take place on Monday, February 27, 5:00 7:00 pm .
Term Test Room Assignments
Surname
go to room
A to C
D to Z
IC 200
IC 13
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Assignment #4
This assignment is due
February 6 February 10, 2017.
at
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Chap
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Assignment #6
The Term Test will take place on Monday, February 27, 5:00 7:00 pm .
Term Test Room Assignments
Surname
go to room
A to C
D to Z
IC 200
IC 13
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Solutions #2
1. f (x) = x (2 x) = 2x x2 on [0, 2].
Since the periodic extension
is even bk = 0,
Z
10
1 2
k = 1, 2, . a0 =
(2x x2 ) dx =
0
3 2
5
1
x
4 2
x2