MATH30010: Field Theory
Homework 2: Solutions
1. Hamiltons quaternions, H, consists of the set of all symbols of the
form a + bi + cj + dk, where a, b, c, d R and i, j, k are symbols
satisfying the id
Chapter 12
Taylor series
12.1
Introduction
The topic of this chapter is find approximations of functions in terms of power series,
also called Taylor series. Such series can be described informally as
Chapter 10
Sequences
10.1
Introduction
This chapter has several important and challenging goals. The first of these is to develop the
notion of convergent and divergent sequences. As an application of
Chapter 5
Applications of the
definite integral to
volume, mass, and
length
5.1
Introduction
In this chapter, we consider applications of the definite integral to calculating geometric
quantities such
Chapter 4
Applications of the
definite integral to
velocities and rates
4.1
Introduction
In this chapter, we encounter a number of applications of the definite integral to practical
problems. We will
Chapter 9
Differential Equations
9.1
Introduction
A differential equation is a relationship between some (unknown) function and one of its
derivatives. Examples of differential equations were encounte
Chapter 3
The Fundamental
Theorem of Calculus
In this chapter we will formulate one of the most important results of calculus, the Fundamental Theorem. This result will link together the notions of an
Chapter 2
Areas
2.1
Areas in the plane
A long-standing problem of integral calculus is how to compute the area of a region in
the plane. This type of geometric problem formed part of the original moti
Chapter 11
Series
11.1
Introduction
This chapter builds on the concept of converging P
and diverging sequences developed in
Chapter 10 to understand when an infinite series j=0 aj converges. Infinite
Chapter 7
Improper integrals
7.1
Introduction
The goal of this chapter is to meaningfully extend our theory of integrals to improper
integrals. There are two types of so-called improper integrals: the
MATH30010: Field Theory
Homework 5: Solutions
1. Let L1 /F and L2 /F be eld extensions. Suppose that : L1 L2
is a eld embedding satisfying (a) = a for all a F (we say that
xes F ). Suppose that b L1
MATH30010: Field Theory
Mid-term Test (2010)
Solutions/Comments
1. Dene what is meant by a eld.
Solution: From class
2. Show that in any eld F , 0 a = 0 for all a F .
Solution: From homework 1.
3. Wha
MATH30010: Field Theory
Homework 3: Solutions
1. Let K be a eld and let F be a subeld (i.e. K/F is an extension of
elds). Let p(x), q(x) F [x] with q(x) = 0. Prove that if q(x) divides
p(x) over K, th
MATH30010: Field Theory
Homework 1: Solutions
1. Prove the uniqueness of multiplicative inverses in any eld F ; i.e., let
a F and let b, c F satisfy
a b = a c = 1.
Using the eld axioms, show that b =
MATH30010: Field Theory
Homework 4: Solutions
1. Let
n := e2i/n = cos
2
n
+ i sin
2
n
C.
(So (n )n = 1.) Explain why n is an element of (multiplicative) order
n in C.
Show that the minimal polynomial
Integral Calculus with Applications to the
Life Sciences
Leah Edelstein-Keshet
Mathematics Department, University of British Columbia, Vancouver
June 24, 2015
Course Notes for Mathematics 103
c Leah K