Table of Contents
Math Bits - Change of Variables
Prerequisites
Students should be able to find derivatives of unspecified functions, especially using the product rule.
In-class Content
- Change the Independent Variable (SWBQ: 15 min) (This was kind of more like a SGA .)
- Change the Dependent Variable (SWBQ: 10 min) (We skipped this in 2017 for time.)
Additional Content
- Reading/Lecture:
QMSHO (Lecture: 25 min)
- A Hermite Polynomial (SGA) 20 min
Student performance
Here are some notes regarding students' performance on the quiz in 2018, which covered change of variables (from Karan, who graded the quiz):
- Since, Weihong mentioned in class that second derivative is not the Square of the first derivative, there were only 2 students who did that.
- There were still a majority ( 8-9) students who zapped the equation TWICE to get d^2 operator.
- Then there were some students ( 4-5) who tried using chain rule and messed up on the second derivative.
- Students who were correctly able to change the first and second derivatives ran out of time/ made algebra mistakes. For those students, I didn't take any points off.
Homework for Central Forces
- (ChangeDerivative)
Consider the differential equation $z^2y^{\prime\prime} + zy^{\prime} - (1-z)y = 0$. (This is known as Bessel's Equation.)
Using the change of variable $u = 2\sqrt{z}$, rewrite Bessel's Equation in terms of $u$.