Motivation for tangent space

The isomorphism between tangents and derivations is surprising! As is the fact it is a linear map. However with calculus one is not far from that definition already.

Motivating example

Let us take (informally, because cases where $\theta=\tfrac{\pi}{2}$ and $r=0$ must be treated carefully) the manifold of the plane. The reader should be familiar with polar coordinates (giving things as an angle and a distance from the origin, rather than $x$ and $y$)

We will have two ways of looking at points, as $(x,y)$ - traditionally, and $(r,\theta)$ where:

• $(r,\theta)\mapsto(r\cos(\theta),r\sin(\theta))$
• $(x,y)\mapsto\left(\sqrt{x^2+y^2},\arctan(\frac{y}{x})\right)$

The line

Take the line $y=mx+c$, where $m$ is the gradient and $c$ is the intercept with the y axis, writing this we see the line can be given as:

Form	First coord	Second coord
$x,y$	$x=t$	$y=mt+c$
$r,\theta$	$r=\sqrt{t^2(m^2+1)+2mct+c^2}$	$\theta=\arctan\left(m+\frac{c}{t}\right)$
Pure forms
Form	map
Cartesian	$y=mx+c$
Polar	$$r=|c|\sqrt{\frac{(m^2+1)}{(\tan(\theta)-m)^2}+\frac{2m}{\tan(\theta)-m}+1}$$

These formulas are easily found from substitution.

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TODO: Add picture, talk about tangents between the two!

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