Watched MIT 18.02 - lecture 11

In 18.02 lecture 11 Prof. Denis Aroux talks about differentials and the Chain Rule. Two of the examples used to illustrate the main topic are of particularly interesting: a new proof for the differentiation of products and quotients, and the conversion between rectangular and polar coordinates.

A main result of this lecture is $$df = f_u \frac{du}{dx} + f_v \frac{dv}{dx},$$ where $f$ is a function of two variables $u,v$ which are both dependent on $x$, and $f_u, f_v$ are partial derivatives. The quotient rule can be derived from this result as follows. Let $g(x) = \frac{u}{v}$, with $u,v$ both dependent on $x$ :
$\begin{align*} df &=f_u \frac{du}{dx}+f_v \frac{dv}{dx} \\ &=  \frac{1}{v}\frac{du}{dx}-\frac{u}{v^2}\frac{dv}{dx} \\ &= \frac{ v \frac{du}{dx}-u \frac{dv}{dx} }{v^2} \end{align*}$
The last expression is the quotient rule for differentiation.

This lecture inspired me to some experimentation ( play ) with Mathematica's PolarPlot function. A polar coordinate is in fact a function of two variables $x,y$ which are both dependent on $r$ and $\theta$ with $x=r \cos(\theta)$, $y=r \sin(\theta)$. By applying the theory above one suddenly gets control over geometric objects like this:

Click to enlarge

Finally the concept of a gradient was mentioned which is merely a vector of partial derivatives. Gradients are the topic of lecture 12. I designed some problems and exercises ( and other experiments ) for functions in polar coordinates. I am delighted I feel more able in that regard.