# Definition of the tangent space and the differential of a smooth function using chart-functions

For an abstract differential manifold $$M$$ of dimension $$m$$, for every point $$p \in M$$, we can define a tangent vector to M at p as a function: $$v:C_p \to \mathbb{R}^m,$$ where $$C_p$$ is the set of all charts of $$M$$ around $$p$$, with the property that for every two charts $$\varphi_1, \varphi_2 \in C_p,$$ we have that $$v(\varphi_2) = (dc)_{\varphi_1(p)}(v(\varphi_1)),$$ where $$c = \varphi_2 \circ \varphi_1^{-1}$$ is the transition function (that is smooth) and $$(dc)_{\varphi_1(p)}$$ is the classical differential of the function $$c$$ in the point $$\varphi_1(p)$$.

Next, we can define that tangent space of M at p as $$T_p(M) = \{v \ \mid \ v \text{ is a tangent vector to } M \text{ at } p \},$$ which has a natural vector space structure over $$\mathbb{R}$$: point-wise addition and multiplication of functions, i.e. $$(v+w)(\varphi):= v(\varphi) + w(\varphi) \text{ and } (c \cdot v)(\varphi):= c \cdot (v(\varphi)),$$ for every $$\varphi \in C_p$$ and every $$v,w \in T_p(M)$$.

So far, this definition seems a bit awkward and difficult to work with, but nevertheless, now I want to define the differential of a smooth function:

For a smooth function $$f:M \to N$$ between two differential manifolds, we define the differential of $$f$$ at $$p \in M$$ as the unique function $$(df)_p : T_p(M) \to T_{f(p)}(N)$$ such that for every $$\varphi \in C_p$$ and for every $$\chi \in C_{f(p)}$$, we have that $$((df)_p(v)) (\chi) = (d(\chi \circ f \circ \varphi^{-1}))_{\varphi(p)}(v(\varphi)), \forall v \in T_p(M),$$ where $$(d(\chi \circ f \circ \varphi^{-1}))_{\varphi(p)}$$ is the classical differential of a function from $$\mathbb{R}^m$$ to $$\mathbb{R}^n$$.

My question is, how are we sure that this $$(df)_p$$ exists (how can we find such a function such that that special property is satisfied) and that it is well defined (that we define this $$(df)_p$$ independently of the charts $$\varphi$$ and $$\chi$$)?

Moreover, how can we prove that this function is actually linear (after we have defined it). I am asking this because, in order to prove that it is linear, I would choose a chart $$\varphi \in C_p$$ and $$v,w \in T_p(M)$$, and then, for every $$\chi \in C_{f(p)}$$, we have that $$((df)_p(v+w))(\chi) = (d(\chi \circ f \circ \varphi^{-1}))_{\varphi(p)}((v+w)(\varphi)),$$ and then linearity follows because the classical differential is a linear function. However, is this proof complete? (I am asking this because I have chosen only a particular chart $$\varphi$$ and proved the linearity, although for every such chart the argument is the same).

There are two separate things to show: (i) you want to show that your definition of $$((df)_p(v))(\chi)$$ is independent of the choice of chart $$\varphi\in C_p$$ and (ii) you want to show that $$(df)_p(v)$$ is a tangent vector in the sense that you've defined it. For (i), suppose $$\varphi_1, \varphi_2 \in C_p$$. Then \begin{align*} (d(\chi\circ f\circ \varphi_2^{-1}))_{\varphi_2(p)}(v(\varphi_2)) &= (d(\chi\circ f\circ \varphi_2^{-1}))_{\varphi_2(p)}((d(\varphi_2\circ\varphi_1^{-1}))_{\varphi_1(p)}(v(\varphi_1)))\\ &= (d(\chi\circ f\circ\varphi_1^{-1}))_{\varphi_1(p)}(v(\varphi_1)) \end{align*} using your definition of a vector in the first equality, and properties of the classical differential in the second equality.
For (ii), suppose $$\chi_1, \chi_2 \in C_{f(p)}$$. Then picking any $$\varphi\in C_p$$, \begin{align*} ((df)_p(v))(\chi_2) &= (d(\chi_2\circ f\circ \varphi^{-1}))_{\varphi(p)}(v(\varphi)) \\ &= (d(\chi_2\circ\chi_1^{-1}))_{\chi_1(p)}((d(\chi_1\circ f\circ \varphi^{-1}))_{\varphi(p)}(v(\varphi))) \\ &= (d(\chi_2\circ\chi_1^{-1}))_{\chi_1(p)} (((df)_p(v))(\chi_1)) \end{align*} Here, the first equality is the definition of the differential, the second uses properties of the classical differential, and the third is again the definition of the differential. Hence $$(df)_p(v)$$ satisfies your criterion for being a tangent vector to $$N$$ at $$f(p)$$.
Now that you've proved $$(df)_p(v)$$ is defined independent of $$\varphi\in C_p$$ and $$\chi\in C_{f(p)}$$, you only need verify linearity in one choice of charts, as you said.