# blueollie

## Taylor series are life saving!

I am getting ready to go swim; I might not run or walk much today. If I do, it will be “5 mileish” (8 km).

Update: 2000 yard swim. What was amusing is that the gasket around my goggles became undone as I took them out; I cursed under my breath and spent about 10 minutes poolside repairing them. A swimmer saw me and invited me to share her lane; that was sweet! 🙂

She thought that I was waiting to have a lane to myself when in fact I was trying to put the gasket back on.

Afterwards I walked home (just over 3 miles) and barely beat the rain; then I did 2 more faster walking miles (12:20 mpm) on the treadmill and some piriformis stretches.

Mathematics: yes, Taylor Series can save lives!

A story about Igor Tamm, the father of tokamak method of controlled thermonuclear fusion:

During the Russian revolution, the mathematical physicist Igor Tamm was seized by anti-communist vigilantes at a village near Odessa where he had gone to barter for food. They suspected he was an anti-Ukranian communist agitator and dragged him off to their leader.

Asked what he did for a living he said that he was a mathematician. The sceptical gang-leader began to finger the bullets and grenades slung around his neck. “All right”, he said, “calculate the error when the Taylor series approximation of a function is truncated after n terms. Do this and you will go free; fail and you will be shot”. Tamm slowly calculated the answer in the dust with his quivering finger. When he had finished the bandit cast his eye over the answer and waved him on his way.

By the way, the answer is (if the series is expanded about 0)
$|f^{(n+1)}(c)|(x)^{(n+1)}/(n+1)!$ where “c” is chosen so that the n+1’st derivative will be at a maximum absolute value in the interval connecting “x” to 0.

Note: you can prove this result by repeated integration by parts.

Note 2: a tokamak is a donut shaped thing that is designed to create a magnetic field to “bottle up” very hot plasma. The idea is that one needs extremely high temperatures for nuclear fusion to take place.

So why is the donut shape necessary?

The reason is that: of all possible surfaces that embed into three space, only the single holed donut shape (called a torus) can have what is called a “vector field which is not zero anywhere). The Klein Bottle has the vector field property, but doesn’t embed into 3-space without intersecting itself.

This is an example of a torus.

If you are curious about what a “vector field” is, I’ll give a quick and dirty oversimplification.

Imagine your surface having lots of lots of hair. Imagine each individual hair being exactly perpendicular to the surface (ok, to the tangent plane of the surface). Now if you shine a light straight over the hair, it casts no shadow at all. The “zero shadow” can be thought of as a “zero vector”. Now bend the hair a bit and shine the light. The shadow of the hair on the surface is called a vector; since there is a shadow we say that the vector is non-zero.

Now, the Gauss-Bonnet theorem says that only the torus (donut) and Klein Bottle have the property that every single hair can be combed so as to produce a shadow; it is impossible to do that on a sphere (called the “hairy ball theorem“).

Here is an attempt to produce a nowhere zero vector field on a sphere; this attempt fails at the poles.

This is an example of the torus with a nowhere zero vector field.