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MATH 127 : Calculus 1 for the Sciences
Weekly Electronic Assignment #10:
Applications of Dierentiation and Introduction to Integration
Due by 9:00 pm on Wednesday, March 16, 2011
Instructions:
Ensure you have completed the weekly tasks up to and includ
Distance Education MATH 127 Calculus 1 for the Sciences
B. A. Forrest
FINAL EXAM REVIEW
OVERVIEW:
This document contains the following:
A brief description of what to expect on the exam.
Recommendations on studying for the nal exam in DE Math 127.
Spec
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MATH 127 : Calculus 1 for the Sciences
Instructor: B. Forrest
Self Check #3: Properties of the Derivative
Recommended: complete by February 19, 2011
Weight: none - self graded - nal exam review
Instructions:
Ensure you have completed the weekly tasks u
Math 127
Calculus 1 for the Sciences
Fall 2011
Objectives:
1. To give a thorough discussion of dierentiation and integration, the basic
processes of calculus, and their application in a wide variety of contexts.
2. To give you experience in writing well-o
UNIVERSITY OF WATERLOO
FINAL EXAM
FALL TERM 2010
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
MATH 127
COURSE TITLE
Calculus 1 for the Sciences
COURSE SECTION(s)
001 002 003 004 005 006 007 008 009
DATE
Chapter 16
Wiener Measure and
Brownian Motion
Diusion of particles is a product of their apparently random motion. The
density u(t, x) of diusing particles satises the diusion equation
u
= u.
t
(16.1)
If the initial condition u(0, x) = f (x) for x R n is
2. Linear transformations and matrices
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2. In the context of Exercise 1, specify N (D), N (I), R(D), and R(I).
3. Consider A, B : R3 R3 ,
0 1
0 0
A=
0 0
given by
0
0
, B = 1
1
0
0
0 0
0 0.
1 0
Compute AB and BA.
4. In the context of Exercise 3, specify
N
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Linear Algebra Michael Taylor
(b) Show that N (A) = 0 if and only if the columns of A are linearly
independent.
10. Dene the transpose of an m n matrix A = (ajk ) to be the n m
matrix At = (akj ). Thus, if A is as in (2.3)(2.4),
a11 am1
.
.
.
.
(2.25)
3. Basis and dimension
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By convention, if V has only one element, the zero element, we say V = 0
and dim V = 0.
It is easy to see that any nite set S = cfw_v1 , . . . , vk V has a maximal
subset that is linearly independent, and such a subset has the s
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Linear Algebra Michael Taylor
where vj = (v1j , . . . , vkj )t . Then the last component of each of the vectors
w1 , . . . , wk is 0, so we can regard these as k vectors in Fk1 . By induction,
there exist scalars a1 , . . . , ak , not all zero, such th
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Linear Algebra Michael Taylor
especially
dim M (n, F) = n2 .
6. If V and W are nite dimensional vector spaces, n = dim V , m =
dim W , what is dim L(V, W )?
Let V be a nite dimensional vector space, with linear subspaces W and
X. Recall the conditions
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Linear Algebra Michael Taylor
Proposition 3.8. Assume V and W are nite dimensional and A : V W
is linear. Then
(3.11)
A surjective AB = IW , for some B L(W, V ),
and
(3.12)
A injective CA = IV , for some C L(W, V ).
Proof. Clearly AB = I A surjective a
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Linear Algebra Michael Taylor
vector in Fk whose jth entry is 1 and whose other entries are 0. First note
that
b
1j
(2.19)
.
Bej = . ,
.
bnj
the jth column in B, as one can see via (2.4). Similarly, if D denotes the
right side of (2.16), Dej is the jth
2. Linear transformations and matrices
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We can also compose linear transformations S L(W, X), T L(V, W ):
(2.11)
ST : V X,
(ST )v = S(T v).
For example, we have
Mf D : C k+1 (I) C k (I),
(2.12)
Mf Dg(x) = f (x)g (x),
given f C k (I). When two transformat
Chapter 15
Probability Spaces
and Random Variables
We have already introduced the notion of a probability space, namely a
measure space (X, F, ) with the property that (X) = 1. Here we look
further at some basic notions and results of probability theory.
Random Fields: Stationarity, Ergodicity, and Spectral Behavior
Michael Taylor
Contents
1. Introduction and denitions
2. Implications of the ergodic theorem
3. Stationary Gaussian elds
4. Ergodic Gaussian elds
5. Stationary random elds on Lie groups
6. Sta
Alternative Formulation of the Wiener Criterion
Michael Taylor
Let be a bounded open set in Rn , K = Rn \ . If p , the Wiener criterion
states that p is a regular point for the Dirichlet problem if and only if
2(n2)j cap(K Bj (p) = ,
(1)
j=0
where Bj (p)
Regularity of Weakly Conformal Maps
Michael Taylor
(In Consultation with Norberto Kerzman)
Let C be an open set. A C 1 map f : C is said to be weakly conformal
provided
(1)
|x f |2 = |y f |2 ,
x f, y f = 0,
on , where , is the standard inner product on R2
Uniformization of Compactly Perturbed Planes,
And Related Green Function Constructions
Preliminary Notes
Michael Taylor
1. Introduction
This work is motivated by an issue in geometrical optics, concerning the null
bicharacteristics of a variable speed dAl
Linear Algebra
Michael Taylor
Contents
1. Vector spaces
2. Linear transformations and matrices
3. Basis and dimension
4. Matrix representation of a linear transformation
5. Determinants and invertibility
6. Eigenvalues and eigenvectors
7. Generalized eige
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Linear Algebra Michael Taylor
F = R or C. In 2 we consider linear transformations between such vector
spaces. In particular we look at an m n matrix A as dening a linear
transformation A : Fn Fm . We dene the range R(T ) and null space
N (T ) of a linea
1. Vector spaces
3
There is also multiplication by scalars; if a is a real number (a scalar),
(1.3)
av = (av1 , . . . , avn ).
We could also use complex numbers, replacing Rn by Cn , and allowing a C
in (1.3). We will use F to denote R or C.
Many other ve
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Linear Algebra Michael Taylor
We give some other examples of vector spaces. Let I = [a, b] denote an
interval in R, and take a non-negative integer k. Then C k (I) denotes the set
of functions f : I F whose derivatives up to order k are continuous. We
d
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Linear Algebra Michael Taylor
is said to be a linear transformation provided
(2.2)
T (a1 v1 + a2 v2 ) = a1 T v1 + a2 T v2 ,
aj F, vj V.
We also write T L(V, W ). In case V = W , we also use the notation
L(V ) = L(V, V ).
Linear transformations arise in
2. Linear transformations and matrices
5
Hint. Feel free to use the results of (1.12).
Let V be a vector space (over F) and W, X V linear subspaces. We
say
(1.16)
V =W +X
provided each v V can be written
(1.17)
v = w + x,
w W, x X.
We say
(1.18)
V =W X
pr
3. Basis and dimension
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Proposition 3.9. Let A be an n n matrix, dening A : Fn Fn . Then
the following are equivalent:
A is invertible,
The columns of A are linearly independent,
(3.21)
The columns of A span Fn .
Exercises
1. Suppose cfw_v1 , . . . , vk
4. Matrix representation of a linear transformation
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Note that the jth column of A consists of the coecients of T vj , when this
is written as a linear combination of w1 , . . . , wm . Compare (2.19).
If we want to record the dependence on the bases S a
3. Basis and dimension
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Proposition 3.6. Assume V and W are vector spaces, V nite dimensional, and
(3.6)
A : V W
a linear map. Then
(3.7)
dim N (A) + dim R(A) = dim V.
Proof. Let cfw_w1 , . . . , w be a basis of N (A) V , and complete it to a basis
cfw
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Linear Algebra Michael Taylor
6. Eigenvalues and eigenvectors
Let T : V V be linear. If there is a nonzero v V such that
(6.1)
T v = j v,
for some j F, we say j is an eigenvalue of T , and v is an eigenvector.
Let E(T, j ) denote the set of vectors v V
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Linear Algebra Michael Taylor
3. Let A M (n, C). We say A is diagonalizable if and only if there exists
an invertible B M( n, C) such that B 1 AB is diagonal:
1
.
.
B 1 AB =
.
n
Show that A is diagonalizable if and only if Cn has a basis of eigenvecto
Name Student ID #
Lil's @kl
Math 127 Calculus for the Sciences I
Midterm Spring 2015
June 15, 2015
3:30 pm 4:50 pm
This test contains 8 pages, including this cover page and a page at the end for rough work.
Write your student ID number at the top of this
Circle
The equation for a circle is given by
(x x0 )2 + (y y0 )2 = r2
The main points to know about circles are:
The centre of the circle is at (x0 , y0 )
The radius of the circle is r.
You should be able to complete the square to re-write an equation
Example 0.1. A box has a base which is twice as wide as it is deep. The volume
of the box is 576 cm3 . What is the smallest possible surface area it can have?
Solution 0.2. Lets let the depth of the box be x and the height be y. The width is
2x, so the vo