Definition:
Let A Mn(R). If a nonzero vector v Rn and
R have the property A v = v then is called an eigenvalue
of A and v is said to be an eigenvector of A associated to the
eigenvalue .
Theorem: If A Mn(R), det A = product of the eigenvalues (counted
wi
Theorem
Let f : Rn R be differentiable and let a be an
interior point in the domain of f . If f has a local extremum at a,
then f (a) = 0.
Definition
Those points a, in the domain of f , at which f is
either not differentiable or D f (a) = O are called cr
Theorem Change of Variables Theorem
Let B, B be regions in Rn and let g : B B be of class C 1 with
g
a bijection onto interior(B) and B = g(B ). Then for any
interior(B )
integrable function f : B R we have
Z
f (x) dVx =
B
Z
f (g(u) | Jg(u) | dVu .
B
Jg(
Theorem: If B is bounded and f is continuous over B, then f is integrable
over B.
Definition The graph of a polar equation (equation in r and ) is the set
of points that corresponds to the polar coordina tes (r, ) that satisfy the
equation.
Theorem Suppos
Definition: Let v = (v1, v2, v3), w = (w1, w2, w3) R3. We
define the cross product v w by
v w = (v2 w3 v3 w2, v3 w1 v1 w3, v1 w2 v2 w1) .
Properties:
1. v w v, w
2. v w = w v
3. v (u + w) = v u + v w
(v + u) w = v w + u w
4. k v w k = kvk kw k | sin |, wh
Definition.
A function, whose partial derivatives exist and are
continuous, is said to be of class C1.
If the partial derivatives of a C1 function f have, in
Definition.
turn, continuous partial derivatives, we say f is of class C2 (twice
continuously dif
To look at the rate of change in a function f at a point a along the
line p + t v, t R we have
Definition:
The directional derivative of f at a in direction
v, denoted by Dv f (a) , is given by
f (a + tv) f (a)
Dv f (a) = lim
.
t0
tkvk
In the special case
Theorem 1. Let f, g : U Rn Rk be differentiable at a Rn
and let R. Then
(i) f is differentiable at a and
D f (a) = D f (a)
(ii) f + g is differentiable at a and
D f + g (a) = D f (a) + D g(a)
Theorem 2. Let f, g : U Rn R be differentiable at a Rn.
Then
(i
We consider 2 practical approaches to computing the triple integral
ZZZ
Z
f dV
f dV =
B
B
1.
Z ZZ
f dA dz
Fix one variable, say (w.l.o.g.) z and let Rz be the planar region consisting of the points in B with z fixed (the cross section).
Z ZZ
Z
f dA dz.
f
Taylor series around a = 0 that you should know.
X
xk
x
e
=
, |x| <
k!
k=0
cos x
sin x
=
X
=
k=0
X
k=0
x2k
(1)
(2k)!
=
ln(1 + x) =
k=0
X
arctan x =
, |x| <
x2k+1
, |x| <
(1)
(2k + 1)!
X
1
1+x
k
(1)k xk
, |x| < 1
k
xk+1
(1)
k+1
, |x| < 1
x2k+1
(1)
2k +
Z
D
(1 + xy) dA, where D = (x, y) R2 | 1 x2 + y 2 2, y 0 .
y=
y=
!
- 2
Z
"#2#
2-x
"#2#
1-x
-1
(1 + xy) dA =
Z
1
1
Z
2x2
Z
1
Z
!
2
2x2
(1 + xy) dy dx +
(1 + xy) dy dx +
2
1x
1
2x2
2x2
Z 1
Z 1
Z
1 2
1 2
(1+xy) dy dx = y + xy
y + xy
dx+
dx+
2
2
2
1
0
1
2
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #2
This assignment is due
January 31 February 2, 2006.
at
the
start
of
your
tutorial
in
the
period
B. Problems:
1. Find the Fourier series of
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #3
This assignment is due
February 7 February 9, 2006.
A. Suggested reading:
at
the
start
of
your
tutorial
in
the
period
1. Marsden & Tromba,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #4
This assignment is due at
February 14 February 16, 2006.
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #5
This assignment is due
February 28 March 2, 2006.
at
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Chap
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #6
This assignment will NOT be collected since the Midterm Test will take place on
Friday, March 3, 7:00 pm 9:00 pm. The midterm test may inc
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #7
This assignment is due
March 14 March 16, 2006.
at
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Chapte
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #8
This assignment is due
March 21 March 23, 2006.
A. Suggested reading:
at
the
start
of
your
tutorial
in
the
period
1. Marsden & Tromba, Cha
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #9
This assignment is due
March 28 March 30, 2006.
at
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Chapte
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #10
This assignment is
April 4 April 6, 2006.
due
at
the
start
of
your
tutorial
in
the
A. Suggested reading:
period
1. Marsden & Tromba, Chap
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #11
This assignment will not be collected. The solution set should be available at the end of the
term.
Problems:
40
1. Consider the counterc
University of Toronto Scarborough Department of Computer & Mathematical Sciences
MAT B42H Assignment #1 This assignment is due January 21 January 24, 2008. B. Problems: 1. Show that
2007/2008
at
the
start
of
your
tutorial
in
the
period
(a)
(b)
sin(kx) cos
University of Toronto Scarborough Department of Computer & Mathematical Sciences
MAT B42H Assignment #2 This assignment is due January 28 January 31, 2008. B. Problems: 1. (a) Find the Fourier series of the sawtooth function f (x) of period 2 whose values
University of Toronto Scarborough Department of Computer & Mathematical Sciences
MAT B42H Assignment #3 This assignment is due February 4 February 7, 2008. A. Suggested reading: 1. 2. 3. B. Problems: 1. A particle following the curve (t) = (sin et , t, 4
University of Toronto Scarborough Department of Computer & Mathematical Sciences
MAT B42H Assignment #4 This assignment is due at February 11 February 14, 2008. the start of your tutorial in the period 2007/2008
A. Suggested Reading: Marsden & Tromba, Cha
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2007/2008
Assignment #5
The Term Test will take place on Friday, March 7, 7:00 pm 9:00 pm in SW128.
This assignment is due at
February 25 February 28, 2008.
the
star
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2007/2008
Assignment #6
The Term Test will take place on Friday, March 7, 7:00 pm 9:00 pm in SW128.
This assignment is
March 3 March 6, 2008.
due
at
the
start
of
you
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2007/2008
Assignment #7
This assignment will NOT be graded since the Term Test will take place on
Friday, March 7, 7:00 pm 9:00 pm in SW 128. The term test will incl
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2007/2008
Assignment #8
This assignment is due
March 17 March 20, 2008.
A. Suggested reading:
at
the
start
of
your
tutorial
in
1.
Marsden & Tromba, Chapter 4, sectio