## Friday, September 17, 2010

### Introductory Maths and Stats: Multi-Variable Calculus...

This is a continuation of my notes for my intro maths and stats tutorials.  This is my summary of Lecture Two: Multi-Variable Calculus...

Partial Differentiation: Easy to extend differentiation from single variable to multi-variable case.  Say you have f(x,y), then to take partial derivative with respect to x simply treat y as a constant and take the derivative of f with respect to x like single variable case!  That's it...also higher order derivatives are calculated by successive application of differentiation.  Demonstrate that cross-partial derivatives are equal (if f is well-behaved)!

I think it would be worthwhile to also mention the Hessian Matrix (matrix of second derivatives).  Talk about special cases when matrix is positive (semi) definite of negative (semi) definite.  Can also use it as an excuse to talk about eigenvalues, eigenvectors, determinants, etc from linear algebra.  Example: f(x,y)=x^2 + y^2...

Total Differentiation and Chain Rules: I totally agree with Yu Fu...one should not try to memorize all of the chain rules related to partial differentiation there are just too many combinations and cases.  Better to focus on understanding the concept of total differentiation and then the difference between independent and intermediate variables.  For example: suppose we have the usual case in economics where f(x(t), y(t)) and t=time.  In this case the independent variable is t, and the dependent variable is f (x and y are only intermediate variables that "filter" the effect of t on f).

Implicit Functions and Differentiation: Just another application of partial differentiation and chain rules...

The Envelope Theorem: Understanding the envelope theorem is key in microeconomic price theory.  Mathematically, the envelope theorem is simply an application of chain rules, total differentiation, and partial differentiation!  No sweat...

Systems of Implicit Functions and Jacobian Determinants: BLAH!  OK, first I think the lecture notes need to be re-ordered so that the lecturer reviews determinants, Cramer's rule etc. BEFORE tackling this section.  Note that Cramer's rule is a REALLY inefficient way to solve a system of linear equations!  For QM0 exam the students may have to compute 3x3 determinant, so they need to know a formula for it...I would go with the Co-factor expansion...

Leibnitz's Rule: This is a cut I think...should be covered in detail by lecturer on Ramsey model in Macroeconomics I...

Integration with Several Variables: Move towards the beginning...this is very straightforward and should probably be talked about right after partial differentiation...

Homogeneous and Homothetic Functions: This is a cut.  Not because it isn't important...it is very important (implications of CRTS and such) but I think that the lecturer should cover these topics in class during term.  There is already too much material in the QM0 lecturers and this would allow for more detailed coverage of other topics...

Linear Dynamic System: If we want to keep this material in course, then we need to do a much better job of teaching eigenvalues, eigenvectors, and matrix diagonalization techniques.  Would recommend moving Appendix on eigenvalues, and eigenvectors into the lecture notes and teaching students how to reach the general solution of a linear dynamic system properly...