# Need help understanding differential of function

I have encountered the term differential/pushforward many times in the literature, although I cannot seem to understand just what is meant by it. I still cannot seem to understand the definition of the differential of a multivalued multivariable function

$$f : \mathbb{R}^n \to \mathbb{R}^m$$

and its generalization to differentiable manifolds. I have seen many definitions of the differential, particularly those for tangent vectors on manifolds and the definition with derivations of functions and one involving the Jacobian matrix, but I cannot understand this or any of them, thus I also cannot understand just what is meant by tangent space on manifolds. What does the differential "really mean" and how to use it? In particular, how is the differential/pushforward related to derivations of functions? Could someone please explain the differential to me and possibly using it to define tangent spaces on manifolds. I am frustrated as I have never understood the differential and lack the proper understanding in tangent spaces. All help is appreciated.

• @MauroALLEGRANZA: thanks I tried the Wikipedia entry did not help, unfortunately. – kroner Jan 5 at 16:22

Let $$f\colon\mathbb{R}\to\mathbb{R}$$ and $$x\in\mathbb{R}$$, then $$f$$ is differentiable at $$x$$ if and only if there exists $$\alpha\in\mathbb{R}$$ such that: $$f(x+h)=f(x)+\alpha h+o(h),$$ when $$\alpha$$ exists, it is unique, denoted by $$f'(x)$$ and called the derivative of $$f$$ at $$x$$.

Proof. Assume that there exists $$\beta\in\mathbb{R}$$ such that $$f(x+h)=f(x)+\beta h+o(y-x)$$, then: $$(\alpha-\beta)(y-x)=o(y-x),$$ whence $$\alpha=\beta$$. $$\Box$$

Remark. Notice that $$h\mapsto\alpha h$$ is a linear map.

Geometrically, $$y=f(x)+f'(x)(y-x)$$ is the best line approximation of the graph of $$f$$ around $$x$$.

Now, let $$f\colon\mathbb{R}^m\to\mathbb{R}^n$$ and $$x\in\mathbb{R}^m$$, generalizing the above definition, $$f$$ is differentiable at $$x$$ if and only if there exists a linear map $$\ell\colon\mathbb{R}^m\to\mathbb{R}^n$$ such that: $$f(x+h)=f(x)+\ell(h)+o(h),$$ when $$\ell$$ exists, it is unique, denoted by $$T_xf$$ and called the differential of $$f$$ at $$x$$.

Proof. Assume that there exists a linear map $$\ell'\colon\mathbb{R}^m\to\mathbb{R}^n$$ such that $$f(x+h)=f(x)+\ell'(h)+o(h)$$, then: $$(\ell-\ell')(h)=o(h).$$ Let $$h\in\mathbb{R}^n\setminus\{0\}$$ and $$t\in\mathbb{R}^*$$, then $$\displaystyle\frac{(\ell-\ell')(th)}{\|th\|}=\frac{(\ell-\ell')(h)}{\|h\|}$$ converges toward $$0$$ when $$t$$ goes to $$0$$, therefore: $$\ell(h)=\ell'(h),$$ and this also holds for $$h=0$$. $$\Box$$

Remark. This definition can be extended to maps defined only on an open set of $$\mathbb{R}^m$$, as $$x+h$$ would fall in this open set for $$h$$ being sufficiently small.

I won't explain how to define the tangent space of a manifold and the tangent map of a differentiable function as it is tedious and quite long. However, the main idea is that a manifold is locally an open set of some $$\mathbb{R}^m$$.

I recommend you have a look at An introduction to Differential Manifolds by J. Lafontaine.