Push forward or differential : is there a link with the differential of a function? My question is really naive but in differential geometry we also call differential the push-forward associated to a function $F : M \rightarrow N$ between two manifolds $M$ and $N$.
But I don't see the link between this map $F_*$ and the usual "differential" of a function.
Is there a reason why we call the push-forward differential like the quantity $df$ or there is absolutely no link between the differential of a function : $df$ and the differential=push-forward.
 A: They're the same idea. If $\varphi:M\rightarrow N$ is a function between smooth manifolds, the differential $d\varphi(x)$ is, at each point, the best linear approximation to $\varphi$.


*

*In real space, we have that for a smooth function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, the differential $df(r, \Delta r)$ is a linear approximation of $f$ such that if you pick a point $r$ and (small) change $\Delta r$ in $\mathbb{R}^m$, you have that the differential is (1) a linear function which (2) approximates the change in the value of the function as you deviate slightly from $r$:


$$ f(r+\Delta r) - f(\Delta r) \approx f^\prime(r)\Delta r = df(r, \Delta r) +\epsilon$$


*In more general smooth manifolds, we have a smooth function $\varphi:M\rightarrow N$, and the push-forward $\varphi_*$ has the same property in that assigns to each point a linear function which approximates the change in the output of the function $\varphi$ as you vary the input.

*But in general smooth manifolds, the tangent space is no longer trivially flat everywhere; the appropriate generalization is that these "small changes" are members of the tangent spaces $TM$ and $TN$ at the appropriate points.

*Hence we can think of the push-forward $\varphi_*(x, dx)$ as an assignment of a linear map to each point $x\in M$ which maps small deviations $dx \in TM_{x}$ to deviations $dy\in TN_{\varphi(x)}$ in a linear way. This definition coincides with the usual definition of function differential for real spaces.
A: 
If $M=\mathbb{R}^m$ and $N=\mathbb{R}^n$, then the pushforward of a map $f\colon M\rightarrow N$ is equal to its usual differential.

Proof. Let $x\in M$ and let $\gamma\colon ]-1,1[\rightarrow M$ be a curve passing through $x$, then one has:
$$f_*(x,\gamma'(0)):=(f\circ\gamma)'(0)=\mathrm{d}_xf\cdot\gamma'(0).$$
Therefore, $f_*(x,\cdot)=\mathrm{d}_xf.$ Whence the result. $\Box$
A: For a function $F\colon\mathbb R^n\to\mathbb R^m$ we have
$$
F_*(x,v)=(F(x),d_xFv)
$$
for each $x\in\mathbb R^n$ and $v\in\mathbb R^m$, where $d_xF$ is the $m\times n$ Jacobian matrix.
So $F_*$ and the differential of $F$ are the same thing, provided that you see the latter as a map between the tangent bundles of $\mathbb R^n$ and $\mathbb R^m$.
