Essential Mathematics for Social Scientists (Revised 23 September 2009)

Objectives:
This course introduces graduate students to the essential mathematics that are utilized in most good social science graduate programs. After completing this course, students should be able to take courses in probability theory and statistics, formal mathematical modeling (choice theoretic as well as systems theory), econometric modeling, and other courses in which basic mathematical concepts are required. Taking this course in advance of applied mathematics courses allows students to interact immediately with the applied subject matter without having to spend a great deal of time struggling with mathematical background material. This course does not involve computing. Students taking subsequent courses with mathematical content will learn various approaches to the use of computers for mathematical and statistical analysis. This course gives these students the background necessary to understand the mathematics behind the programming.

Content:
This course covers two essential areas: (1) an introduction to calculus and its applications, and (2) linear algebra (including matrix manipulations). There are no prerequisites for this course. This course is designed to complement other more advanced applied courses with mathematical content.

Course Structure:
This course is currently structured as a seven day course involving morning classes of three hours each. Students are expected to complete course homework during the afternoon and evening hours. During the first five meetings, we cover mathematical notation as well as differential and integral calculus. The remaining two meetings are devoted to linear algebra with calculus applications.

The reading assignments listed below in the course outline are required of all students. Additional suggested reading assignments will be given as the course proceeds, and these readings will focus on applications of the methods covered in the core text. All students are recommended to work together, sharing information and discoveries.

Class Requirements:

Regular reading and writing assignments are matched with class discussions. The course grade depends on the evaluation of all homework assignments, tests, and attendance. The course grades are used only within the Department of Political Science to guide your future study; they are not sent to the Registrar or entered into your college transcript. There are two one-hour exams. All exams are given during class.

The grades are determined as follows:
20% Attendance
20% Homework assignments
30% Midterm Exam
30% Final Exam

The Honor Code is strictly enforced in this course. Plagiarism is an honor code violation. A signature forgery on attendance is an honor code violation.

Podcast Policy:

Podcasting courses can assist students tremendously. Students can listen to lectures more than once, and they can catch up on classes that were missed for, say, reasons of illness or religious obligation. I record and podcast many of the classes in this course. By taking this course, all students are automatically giving their permission to be recorded during class participation. No further written permission is required.

Disabilities Statement:

It is the policy of Emory University to make reasonable accommodations for qualified students with disabilities. All students with special requests or need for accommodations should make this request in person as soon as possible after first visiting the Office of Disabilities.

Required Texts:

Essential Mathematics for Political and Social Research, by Jeff Gill. 2006. Cambridge: Cambridge University Press. ISBN: 978-0521684033.

Basic Math for Social Scientists: Concepts, by Timothy M. Hagle. Thousand Oaks, California: Sage. Number 108 in the Quantitative Applications in the Social Sciences series. ISBN: 978-0803958753.

Basic Math for Social Scientists: Problems and Solutions, by Timothy M. Hagle. Thousand Oaks, California: Sage. Number 109 in the Quantitative Applications in the Social Sciences series. ISBN: 978-0803972858.

Calculus, by Gudmund R. Iversen. Thousand Oaks, California: Sage. Number 110 in the Quantitative Applications in the Social Sciences series. ISBN: 978-0803971103.

INTERNET RESOURCES:

General Math -
A great web site on fractal geometry and chaos
Professor Strang's Linear Algebra Class Lecture Videos (MIT)
Professor Strang's Web Programs for Linear Algebra (MIT)

Data and Statistical -
Emory University Library's Political Methodology Research Guide
Emory University's Electronic Data Center
The Cran Home Page: This is where you get R
Wikipedia's discussion of the R programming language
A discussion in a political methodology journal about the use of R
John Fox's methods class page
The Quick-R page, a great resource for SAS and SPSS users
The "Kickstarting R" Intro Manual
Hugh C. Pumphrey's course notes on R
Thomas Lumley's course notes on R
Rob Cribbie's course notes on R
Frank McCown's easy intro to graphing using R
The R Graphical Manual: A really comprehensive collection of graphics methods
The American Phytopathological Society has published a great introduction to R
Gary King's excellent advice on writing your first publishable paper
How to Use a Codebook, from Princeton University
Inter-University Consortium for Political and Social Research (ICPSR)
National Election Studies, The University of Michigan
Rice Virtual Lab in Statistics
SAS Documentation for version 8.2
SAS Documentation for version 9.1 (This is the one we use in class.)
Statistical Abstract of the United States
Statistics Calculators from UCLA
SticiGui Online Statistics Text, by Philip B. Stark, University of California, Berkeley
Surf Stat, an online statistics text

DAILY OUTLINE FOR CALCULUS AND LINEAR ALGEBRA

Day 1: Mathematical Notation and Calculus
Lectures: Notation and algebra
i. Exponents
ii. Logarithms
iii. Relations
iv. Functions
v. Sets
vi. Permutations and combinations
vii. Basics of analytic geometry
viii. An introduction to calculus
Readings: Gill, chapters 1 and 2; Hagle (concepts), chapter 1, Hagle (problems and solutions), TBA
Written Assignment:

Day 2: Calculus
Lectures: Limits and Differential Calculus
i. Limits
ii. Definition of a derivative
iii. Properties of limits
iv. Continuity
v. Rules of differentiation
vi. Analogies between the rules for differential calculus and finite differences
vii. Minima and maxima
viii. Geometric interpretations of differentiation
Readings: Gill, chapter 5, Hagle (concepts), chapter 2, Hagle (problems and solutions), TBA; Iversen, TBA
Written Assignment: TBA

Day 3: Calculus
Lectures: Differentiation
i. Basic social science applications to differentiation
ii. Working with transcendental functions
iii. Examples of simple first-order differential equations
iv. Newton’s method for finding roots of an equation
v. Working with multivariate functions
vi. Extrema of multivariate functions
vii. Optimization using least squares
viii. Optimization using Lagrange multipliers
Readings: Gill, finish chapter 5; Iversen (TBA); Hagle (concepts), chapters 3&4; Hagle (problems and solutions), TBA
Written Assignment: TBA

Day 4: Calculus
Lectures: Integral Calculus
i. Basic concepts of integration
ii. The indefinite integral
iii. The definite integral
iv. Working with areas under a curve
v. Integrating elementary functions
Readings: Iversen (TBA); Gill (review chapter 5); Hagle (concepts), chapter 5, Hagle (problems and solutions), TBA
Written Assignment: TBA

Day 5: Calculus
Lectures: Rules of Integration
i. Partial fractions
ii. Integration by substitution
iii. Integration by parts
iv. Taylor’s approximation
Readings: Review yesterday’s readings
Written Assignment: TBA

Day 6: Linear Algebra
Lectures: Linear Algebra Basics
i. Vectors
ii. Matrices
iii. Elementary properties of vectors and matrices
iv. Working with matrices
v. Matrix manipulations
vi. Matrix rules
Readings: Gill, chapter 3
Written Assignment: TBA

Day 7: Linear Algebra
Lectures: Applied Use of Matrices, with Calculus Applications
i. Determinants
ii. Inverses
iii. Cramer’s Rule
iv. Eigenvalues and eigenvectors
v. Applying calculus with linear algebra
Readings: Gill, chapters 4, 6; Hagle (concepts), chapter 6, Hagle (problems and solutions), TBA
Written Assignment: TBA

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