# Some Resources for Advanced Math in Economics

I am now in my first semester of my first year of an economics Ph.D. at GMU. The professors wasted no time throwing the math our way. This article describes the kind of math used in Ph.D. economics courses as well as a few good sources for learning.

Topics and places to learn:

• Stats
• No particular sources referenced. So far we have only used STATA, but my understanding is that other software is used by various places.
• Calculus
• Khan Academy has you covered.
• Differential Equations
• This video does a great overview of solution techniques using the mnemonic SHIELDS.
• Once again, Khan Academy. Be sure to do the exercises as well until you can get several in a row correct.
• This article does a good job outlining a particular advanced solving technique using the integrating factor.
• Linear Algebra
• This short video from MIT OpenCourseWare does a good job giving some quick solving techniques for eigenvectors, but it isn’t a comprehensive lesson.
• Khan Academy does a great job here as well, but they currently have only videos without practice exercises. In particular, view the “Eigen-everything” section. They also don’t teach on phase diagrams which you should know.
• For some great differential equation intuition, more on phase diagrams, and a bit on the rigorous formulation of coupled systems and more: Eigenvalues, Eigenvectors, and Differential Equations
• Paul’s Online Math Notes might help as well. I basically just like the phase diagrams again here.
• This paper does a good job reviewing a few special cases of eigenvectors including the identity, projection, and reflection vectors.

Tools to help solve and/or check problems include:

• Wolfram Mathematica. Check with your university as most will allow students free access.
• This StackExchange article describes how to graph a phase diagram in Mathematica!
• Here’s the function to obtain the phase diagram of the identity vector, by yours truly:
• StreamPlot[{E^y, E^x}, {x, -5, 5}, {y, -5, 5}, AspectRatio -> 1]
• Wolfram Alpha. Unfortunately this is a separate product from Mathematica. The free online tool gives the answer but you have to pay for a step-by-step walkthrough.
• This online integral calculator and this online derivative calculator from the same cool peeps. All done in Javascript if you are interested in some advanced mathy JS as I am.

Another tip is that you might want to learn the greek alphabet as well because those symbols are used as frequently or more than English letters in mathy economics. I kid you not one of my professors gave a quiz on these letters.

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