# Tangent space and Jacobian

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.

• I don't know this book. Does every manifold $M$ sit inside $\Bbb R^n$ for some $n$? If so, $TM_x$ will be an $m$-dimensional subspace of $\Bbb R^n$, not necessarily a hyperplane — it will be a hyperplane only when $n=m+1$. The sentence "translating ... one obtains $df_x$" is very sloppy and not informative. Does Willard not explain what $df_x$ is in great detail? At any rate, $df_x$ should map $TM_x$ to $TN_{f(x)}$, when $f\colon M\to N$. Commented Jun 1, 2019 at 18:45
• Commented Jun 1, 2019 at 19:02

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.