Calculating Taylor polynomials for a function in $\mathbb{R}^3$. Hello how does one apply the Taylor polynomials in a function of three variables?
If we consider the function $f: \mathbb{R}^3$ to $\mathbb{R}$ with $(x_1, x_2, x_3)$ mapped to $\sin(x^2_1) + \exp(x_2) + \cos(x_1x_3)$.
How can one calculate the Taylor polynomials of order one, two and three evaluated at the point $x = 0$?
Can someone please give me an idea as to how one tackles it.
 A: Computationally simplest is using the known Taylor expansions of the functions $\sin$, $\exp$, and $\cos$. This would give
$$\eqalign{\sin(x_1^2)&=x_1^2-{1\over6}x_1^6 +{1\over120}x_1^{10}-\ldots\ ,\cr
\exp(x_2)&=1+x_2+{1\over 2}x_2^2+{1\over 6}x_2^3 +\ldots\ ,\cr
\cos(x_1x_3)&=1-{1\over2}x_1^2x_3^2 +{1\over24}x_1^4x_3^4 -\ldots\ .\cr}$$
Now collect all terms up to the desired order. E.g., the third order taylor expansion of $f$ at ${\bf 0}=(0,0,0)$ is given by
$$f(x_1,x_2,x_3)=2+x_2+x_1^2+{1\over2}x_2^2+{1\over6}x_2^3+ R_3({\bf x})\ ,$$
where the remainder term is $o(|{\bf x}|^3)$ when ${\bf x}\to{\bf 0}$.  
Of course there is also a general procedure for the Taylor expansion in several variables. When $f:{\mathbb R}^n\to{\mathbb R}$ is sufficiently smooth then at any point ${\bf p}$ it has differentials of order $0$, $1$, $2$, $3$, etc.. The differential of order $r$ at ${\bf p}$, denoted by $d^r f({\bf p})$, is a homogeneous polynomial in the auxiliary variable ${\bf X}=(X_1, \ldots, X_n)$ and is  given by
$$d^r f({\bf p}).{\bf X}=\sum_{k_1,\ldots, k_r} f_{.k_1\ldots k_r}({\bf p}) X_{k_1}\cdot\ldots\cdot X_{k_r}\ .$$
Here the summation variables $k_j$ run independently from $1$ to $n$, so formally there are $n^r$ terms in this sum. The zeroth differential is just the constant function with value $f({\bf p)}$, and $d^1f({\bf p}).{\bf X}=f_{.1}X_1+\ldots+f_{.n}X_n$ (the $f_{.k}$ evaluated at ${\bf p}$) is the usual "differential" of $f$ at ${\bf p}$.
The Taylor expansion of $f$ at ${\bf p}$ can then be written as
$$f({\bf p}+{\bf X})=\sum_{r=0}^N{1\over r!}d^r f({\bf p}).{\bf X} + R_N$$
where $R_N$ denotes the remainder term. The Taylor theorem says, e.g., that $R_N=o(|{\bf X}|^N)$ when ${\bf X}\to{\bf 0}$.
When ${\bf p}={\bf 0}$ we can replace ${\bf X}=(X_1,\ldots ,X_n)$ by ${\bf x}=(x_1,\ldots,x_n)$ and simply write $f({\bf x})=\sum_{r=0}^N\ldots + R_N$.
