“Real mathematics must be justified as art if it can be justified at all.”
― G.H. Hardy, A Mathematician's Apology
In case of doubts: Why even math for Economics?
"The advancement and perfection of Mathematics are intimately connected with the prosperity of the State."
― Napoleon Bonaparte
- Mathematics for Economists is THE introductory textbook for most topics at hand.
- Wolfram Alpha has you covered for calculus, max/min problems, plots, limits etc...
- Wolfram Mathworld
- Encyclopedia of Mathematics
- MIT OpenCourseWare has virtually unlimited resources for math even beyond the scope of this course.
- The World of Mathematical Reality by Paul Lockhart, if you want a small taste of why math-people like math in itself - and maybe you should too!
- Understanding Mathematics, a study guide by Peter Alfeld, in case you feel lost.
- 3blue1brown, for exceptionally accessible videos on a wide range of topics.
- Understanding Mathematics, for more resources on the basics of understanding mathematics and its peculiarities.
Cores: sets, functions, relations, proofs and proof methods.
For a precise introductory treatment:
- Basic Concepts of Mathematics by Elias Zakon
For a more intensive discussion:
- An Introduction to Set Theory by William A. R. Weiss
As a side note, know that E700 covers a number of topics and concepts which cast a wide web of mathematical tools ― many of which we must posses to understand modern economic theory and practice in detail. Thus all of the following resources have some degree of "intersection": you will find topics from one part also presented in a source listed for another part.
Use whichever resource you consider best for your learning. This said, the material seen in class should always be the reference point.
Cores: metric & normed spaces, convergence of sequences, continuity, compactness of sets.
These topics fall mostly under the umbrella of topology, and you can take a look at:
-
Understanding Analysis by Stephen Abbott - Chapters 2, 3, 4, is the most accessible resource for this Part.
-
Principles of Mathematical Analysis by Walter Rudin - Chapters 1, 2, 3, 4.
-
General Topology - Chapters 1, 2, 3.
-
Modern Real Analysis - Chapter 3.
-
Real Analysis I - Chapters 2, 3.
For a more general overview (YouTube):
- Abstract Vector Spaces by 3blue1brown.
Cores: vectors, matrix algebra, linear transforms, eigenvalues.
Rigorous treatment of linear algebra (vectors, matrices and related things) can be found in:
- Linear Algebra by Peter Petersen - Chapters 1, 2.
- Linear Algebra by Jim Hefferon, a very complete book.
However because linear algebra is not always intuitive, especially in high dimensions, these resources are more accessible:
- Linear Algebra Done Wrong
- Immersive Linear Algebra, an interactive web-book entirely on linear algebra.
Videos (YouTube):
- Essence of Linear Algebra by 3blue1brown.
Cores: multivariable differentiation, constrained/unconstrained optimization.
Regarding differentiation (this topic is also easily covered on Wikipedia, e.g.) take a look at:
-
Principles of Mathematical Analysis by Walter Rudin - Chapter 5.
-
The Calculus of Functions of Several Variables - Chapters 2, 3.
-
Mathematics for Economists - Part III.
While for optimization (in Economics we usually consider a pretty specific subset of optimization techniques) it is best to follow:
- Mathematics for Economists - Part IV.
Lastly, MIT Professor Gilbert Strang recorded masterful lectures on calculus, and you can find them (and the accompanying book) on MIT OpenCourseware:
Two topics that are not covered in E700 are measure theory and integration. By and large these are essential to understand most equations found in probability theory, statistics, macro and micro theory, and many more subfields. If you have a minimum of comfort with things like
then you should be fine.
However, good resources on these topics are:
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Understanding Analysis by Stephen Abbott - Chapters 6, 7, for a simple introduction to the Riemann integral.
-
Principles of Mathematical Analysis by Walter Rudin - Chapters 6, 11.
-
An introduction to measure theory by Terence Tao, one of the great contemporary mathematicians.
Note that when you learn measure theory, you also learn Lebesgue–Stieltjes integration, which is the bedrock of modern probability theory - a skill you'll use forever!
Most links are taken from: https://github.com/rossant/awesome-math
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