1
MATH 127 : Calculus 1 for the Sciences
Weekly Online Assignment #8: Higher Derivatives and the Mean Value Theorem
Due by 9:00 pm on WEDNESDAY, March 2, 2016
Weight: 2%
Instructions:
Ensure you have completed the weekly tasks up to and including Monday
1
MATH 127 : Calculus 1 for the Sciences
Weekly Electronic Assignment #1
Due by 9:00 pm on THURSDAY, January 6, 2011
Instructions:
Ensure you have completed A Short Introduction to Maple in the Daily Tasks and read Section 1.1 What is
Calculus? in the Co
1
MATH 127 : Calculus 1 for the Sciences
Weekly Online Assignment #3: Introduction to Sequences
Due by 9:00 pm on WEDNESDAY, January 20, 2016
Weight: 2%
Instructions:
Ensure you have completed the weekly tasks up to and including Thursday, Week 2, Sequen
Assignment 2
Complete this assignment after you have finished Unit 3, and submit your work to your tutor for
grading.
This assignment has one bonus question. There is no penalty if you do not attempt it but you
may be rewarded if you do. The maximum grade
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DE MATH 127 : Calculus 1 for the Sciences
Instructor: B. Forrest
INDEPENDENT TERM PROJECT
Weight: 10%
Due: RECEIVED at the Centre for Extended Learning (CEL) Office or UW Campus Drop Box
before NOON (Waterloo, Ontario time) on MONDAY, November 28, 2016
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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
1
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
1
MATH 127 : Calculus 1 for the Sciences
Instructor: B. Forrest
Self Check #4: Mean Value Theorem and Applications of Dierentiation
Recommended: complete by March 12, 2011
Weight: none - self graded - nal exam review
Instructions:
Ensure you have complet
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
8
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
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
2
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
4
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
2. Linear transformations and matrices
7
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
6
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
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
Regularity for a Class of Elliptic Operators with Dini
Continuous Coecients
Michael Taylor
Abstract
We obtain regularity results for solutions to P u = f when P is a kth order elliptic
dierential operator with the property that both P and P t have coecien
11
Brownian Motion and Potential
Theory
Introduction
Diusion can be understood on several levels. The study of diusion on a
macroscopic level, of a substance such as heat, involves the notion of the
ux of the quantity. If u(t, x) measures the intensity of
Chapter 14
Ergodic Theory
Throughout this chapter we assume (X, F, ) is a probability space, i.e., a
measure space with (X) = 1. Ergodic theory studies properties of measurepreserving mappings : X X. That is, we assume
(14.1)
S F = 1 (S) F and (1 (S) = (S
ELECTROSTATIC SCREENING
Jeffrey Rauch and Michael Taylor
Department of Mathematics
The University of Michigan
Ann Arbor, Michigan 48104
Abstract. Using the methods of partial dierential equatiions and functional
analysis, we investigate the electromagneti
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
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
Lvy Processes
e
Michael Taylor
1. Introduction
Wiener measure is a measure on the space of paths in Rn having the following
property. Consider the Gaussian probability distribution
p(t, x) = (2)n
(1.1)
2
et| eix d
2
= (4t)n/2 e|x|
/4t
.
Given 0 < t1 < t2
2. Linear transformations and matrices
9
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