## Differentiation done correctly: 2. Higher derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Last time, we covered ...

## Differentiation done correctly: 1. The derivative

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In multivariable calcu...

## Some series convergence problems

Here are some series convergence problems that I gathered quite a while ago. A few of them are a bit tricky.

## Every continuous open mapping of R into R is monotonic

I’m willing to bet that most students who have used Rudin’s Principles of Mathematical Analysis have encountered this problem: 15. Call a mapping of...

## Free product of free groups and group presentations

Here is Problem 9-4(b) from Introduction to Topological Manifolds by John M. Lee: Let \(S_1\) and \(S_2\) be disjoint sets, and let \(R_i\) be a subset of the f...

## The “both open and closed” trick for connected spaces

Here is a basic result concerning connected topological spaces. The only subsets of a connected space \(X\) that are both open and closed are \(\emptyset\) and ...

## Formula for the circumference of an ellipse

It is well known that a circle with radius r has diameter 2πr. But is there a similar formula for the circumference of an ellipse?

## Strictly positive extensions of linear functionals

There is an interesting exercise in Steven Roman’s book Advanced Linear Algebra:

## How many triangles in a triangle?

I hope most of you will be familiar with the “how many triangles” puzzle. If you aren’t, here’s a nice demonstration for you.

## Power series of tan(x), cot(x), csc(x)

Here’s a little how-to on figuring out the power series of tan(x), cot(x) and csc(x). Start with the generating function for the Bernoulli numbers: