Proof of Clairaut’s theorem in Terence Tao Analysis 2

I'm self-studying the book Real Analysis 2 by Terence Tao.

I'm having some trouble working with vector-valued functions. Particularly, I'm stuck at his proof of Clairaut's theorem for a vector-valued function while having no math professor to ask :(

screenshot 1 Theorem 6.5.4 (Clairaut's theorem). Let $E$ be an open subset of $\mathbf{R}^{n}$, and let $f: E \rightarrow \mathbf{R}^{m}$ be a twice continuously differentiable function on E. Then we have $\frac{\partial}{\partial x_{j}} \frac{\partial f}{\partial x_{i}}\left(x_{0}\right)=\frac{\partial}{\partial x_{i}} \frac{\partial f}{\partial x_{j}}\left(x_{0}\right)$ for all $1 \leq i, j \leq n$.

In his proof, he proves for the case $\fbox{m=1}$ and $x_0 = 0$. The beginning of the proof is easily to understood:

screenshot 2 Let $a$ be the number $a:=\frac{\partial}{\partial x_{j}} \frac{\partial f}{\partial x_{i}}(0),$ and $a^{\prime}$ denote the quantity $a^{\prime}:=\frac{\partial}{\partial x_{i}} \frac{\partial f}{\partial x_{j}}(0) .$ Our task is to show that $a^{\prime}=a$. Let $\varepsilon>0$. Because the double derivatives of $f$ are continuous, we can find a $\delta>0$ such that $$ \left|\frac{\partial}{\partial x_{j}} \frac{\partial f}{\partial x_{i}}(x)-a\right| \leq \varepsilon $$ and $$ \left|\frac{\partial}{\partial x_{i}} \frac{\partial f}{\partial x_{j}}(x)-a^{\prime}\right| \leq \varepsilon $$ whenever $|x| \leq 2 \delta$.

Then comes the difficult part that I can't understand his notation:

screenshot 3 Now we consider the quantity $$ X:=f\left(\delta e_{i}+\delta e_{j}\right)-f\left(\delta e_{i}\right)-f\left(\delta e_{j}\right)+f(0) $$ From the fundamental theorem of calculus in the $e_{i}$ variable, we have $$ f\left(\delta e_{i}+\delta e_{j}\right)-f\left(\delta e_{j}\right)=\int_{0}^{\delta} \frac{\partial f}{\partial x_{i}}\left(x_{i} e_{i}+\delta e_{j}\right) d x_{i} \qquad\color{red}{(*)}$$ and $$ f\left(\delta e_{i}\right)-f(0)=\int_{0}^{\delta} \frac{\partial f}{\partial x_{i}}\left(x_{i} e_{i}\right) d x_{i} $$ and hence $$ X=\int_{0}^{\delta}\left(\frac{\partial f}{\partial x_{i}}\left(x_{i} e_{i}+\delta e_{j}\right)-\frac{\partial f}{\partial x_{i}}\left(x_{i} e_{i}\right)\right) d x_{i}. $$

Where $e_i$ is a $n-dimensional$ vector with all zeros except the $1$ in the $i^{th}$ position.

My question concerns the $\color{red}{(*)}$ equations:

1. How to use the Fundamental theorem of calculus to lead to the (*) equation?

2. Why the limit of the integral is from $0$ $\to$ $\delta$ and NOT from ($\delta \, e_j$ $\to$ $\delta \, e_i + \delta \, e_j$) ? (like in the case of single variable integration: $F(b) - F(a) = \int \limits_{(a,b)} f $ ) ?
   ($i.e$ In my opinion, we consider $b = \delta \, . e_i + \delta \, .e_j$ and $a= \delta \, .e_j$, so the limit of the integration should be from $a$ to $b$ which is different from what is written in the text: $0$ $\to$ $\delta$)

3. Why the "main" function to integrate is $\frac{\partial f}{\partial x_j}$? Why is it evaluated at the point ($x_i*e_i + \delta * e_j$)?

4. Why the differential part in the integral is $dx_i$?

5. Could you please suggest to me a $\fbox{document}$ or some $\fbox{keywords}$ so that I can read more to further understand the things related to the $\color{red}{(*)}$ equation?

6. In the case where $m > 1$, does the equation $\color{red}{(*)}$ still hold ? Is there any definition of integration of a vector-valued function?
   ($i.e.$ $\int f dx$ where $f : \mathbb{R^n} \to \mathbb{R}^m$ with $m > 1$ and $x$ is a single variable. In other words, does an integration always take scalar- value or it can be a vector) ?

Thank you very much for your help!

Let me share following thoughts:

1. Fundamental theorem of calculus leads to the equation, because (*), and equality after it, is exactly fundamental theorem applied to specific functions and bounds.

2. Antiderivative of $\frac{\partial f}{\partial x_i}(x_i e_i + \delta e_j)$ with respect to $x_i$ is $f(x_i e_i + \delta e_j) +C$, for some $C$ constant and it is used for bounds $0$ and $\delta$ and gives by fundamental theorem $f(x_i e_i + \delta e_j)\big|_{x_i=0}^{x_i=\delta} = f( \delta e_i + \delta e_j) - f( \delta e_j)$.

3. and 4. have same answer: it is proved, in equality (*) and equality after it, that quantity $X$ can be represented in integral form.