I'm reading John Willards Topology with a differential view point and an confused about tangent spaces.
To define the notion of derivative $df_x$ for a smooth map between smooth manifolds we introduce a tanget space at each point $x$ in the manifold $M$. The tangent space is denoted $TM_x$. If $M$ is an $m$-dimensional manifold then $TM_x$ is the $m$-dimensional hyperplane through the origin parallel to the hyperplane that that best approximates $M$ at $x$. Similarly one things of the nonhomegeneous linear mapping from the tangent hyperplane at $x$ to the tangent hyperplane at $y$ which best approximates $f$.
My confusion lies the following sentence: Translating both hyperplanes to the origin, one obtains $df_x$.
Is this saying $df_x$ is a map between these two hyperplanes? If so, how should I think about this map what is getting mapped to what?
Edit: All manifolds are in $R^n$ for some $n$, but the two manifolds may not be of the same dimension.
Best Answer
I hope this helps you clarify this important concept of the differential map. Please note that I deliberately avoid speaking of hyperplanes.
Let $f:M\to N$ be a map between two manifolds of possibly different dimension. The differential map does the following, point-wise: at each point $p\in M$, it gives us a map $$df_p:T_pM\to T_{f(p)}N;\quad v\mapsto df_p(v). $$
Think of $M$ as a manifold of its own, without any space around it. Do the same for $N$. Now, remember that any point $p\in M$ has a coordinate chart that maps a neighbourhood $U$ of $p$ to an open subset of $\mathbb R^{\dim M}$. This is just a mathematical way to say that $p$ is surrounded by something that looks like $\mathbb R^{\dim M}$, as are all points in $M$.
Since $M$ looks like $\mathbb R^{\dim M}$, there are as many as $\dim M$ directions near $p$ that are linearly independent. This gives us that the tangent space of $M$ at $p$, $T_p M$ is a vector space whose dimension matches that of $M$. You can think of it as a copy of $\mathbb R^{\dim M}$ now. Do the same for $N$.
The differential map $df_p:T_p M\to T_{f(p)} N$ is the natural relation induced by $f$ on these tangent spaces. Roughly speaking, it associates to each direction in $T_p M$ the direction in $T_{f(p)}N$ that will $f(p)$ following as a result.
If you want an example, think of $M$ as the 2-sphere with what I am calling terrestrial coordinates, i.e. $(\alpha, \theta)$, where $\alpha$ is the angle of the meridian passing through a point with a fixed meridian, and $\theta$ is the inclination of the parallel containing the point (these are the coordinates you find on any map or GPS). Think now of $N$ as the circle with the single coordinate $\rho$. In these coordinates, we define a map $$f(\alpha,\theta) = \alpha+\rho_0, $$ where $\rho_0$ is a constant we have determined. Intuitively, $f$ forgets about the latitude and then pushes the point forward by $\rho_0$ in the direction of $\alpha$.
The differential map is $df_p = (1,0)$. This means that $df_p(v) = (1,0)(v)$ for any $v\in T_p M$. Indeed, this map varies as fast as $\alpha$ does.