Added Taylor polynomials.

docs/en/mathematics/multivariable-calculus/functions-of-several-variables.md

@@ -19,3 +19,56 @@
 
 *Definition*: let $D \subseteq \mathbb{R}^2$ and let $f: D \to \mathbb{R}$ then for $c \in \mathbb{R}$ we have $S_c := \big\{(x, y) \in D \;\big|\; f(x,y) = c \big\}$ is the level set of $f$. Observe that $S_c \subseteq \mathbb{R}^2$. 
 
+## Multi-index notation
+
+*Definition*: an $n$-dimensional multi-index is an $n$-tuple of non-negative integers
+
+$$
+    \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n), \qquad \text{with } \alpha_i \in \mathbb{N}.
+$$
+
+### Properties
+
+For the sum of components we have: $|\alpha| := \alpha_1 + \dotsc + \alpha_n$. 
+
+For $n$-dimensional multi-indeces $\alpha, \beta$ we have componentwise sum and difference
+
+$$
+    \alpha \pm \beta := (\alpha_1 \pm \beta_1, \dotsc, \alpha_n \pm \beta_n).
+$$
+
+For the products of powers with $\mathbf{x} \in \mathbb{R}^n$ we have
+
+$$
+    \mathbf{x}^\alpha := x_1^{\alpha_1} x_2^{\alpha_2} \dotsc x_n^{\alpha_n}.
+$$
+
+For factorials we have
+
+$$
+    \alpha ! = \alpha_1 ! \cdot \alpha_2 ! \cdots \alpha_n !
+$$
+
+For the binomial coefficient we have
+
+$$
+    \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} \alpha_1 \\ \beta_1 \end{pmatrix} \begin{pmatrix} \alpha_2 \\ \beta_2 \end{pmatrix} \cdots \begin{pmatrix} \alpha_n \\ \beta_n \end{pmatrix} = \frac{\alpha !}{\beta ! (\alpha - \beta)!}
+$$
+
+For polynomials of degree less or equal to $m$ we have
+
+$$
+    p(\mathbf{x}) = \sum_{|\alpha| \leq m} c_\alpha \mathbf{x}^\alpha,
+$$
+
+as an example for $m=2$ and $n=2$ we have
+
+$$
+    p(\mathbf{x}) = c_1 + c_2 x_1 + c_3 x_2 + c_4 x_1 x_2 + c_5 x_1 ^2 + c_6 x_2^2 \qquad c_{1,2,3,4,5,6} \in \mathbb{R}
+$$
+
+For partial derivatives of $f: \mathbb{R}^n \to \mathbb{R}$ we have
+
+$$
+    \partial^\alpha f(\mathbf{x}) = \partial^{\alpha_1}_{x_1} \dotsc \partial^{\alpha_n}_{x_n} f(\mathbf{x}).
+$$

docs/en/mathematics/multivariable-calculus/implicit-equations.md

@@ -1,6 +1,6 @@
 # Implicit equations
 
-*Theorem*: for $D \subseteq \mathbb{R}^2$ (for simplicty), let $f: D \to \mathbb{R}$ be continuously differentiable and $\mathbf{a} \in D$.  Assume
+*Theorem*: for $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicty), let $f: D \to \mathbb{R}$ be continuously differentiable and $\mathbf{a} \in D$.  Assume
 
 * $f(\mathbf{a}) = 0$, 
 * $\partial_2 f(\mathbf{a}) \neq 0$, nondegeneracy. 

docs/en/mathematics/multivariable-calculus/taylor-polynomials.md

@@ -0,0 +1,37 @@
+# Taylor polynomials
+
+For $D \subseteq \mathbb{R}^n$ let $f: D \to \mathbb{R}$ sufficiently often differentiable, we have $\mathbf{a} \in D$. Find a polynomial $T: \mathbb{R}^n \to \mathbb{R}$ such that
+
+$$
+    \partial^\beta T(\mathbf{a}) = \partial^\beta f(\mathbf{a}).
+$$
+
+Ansatz: let $T(\mathbf{x}) = \sum_{|\alpha| \leq n} c_\alpha (\mathbf{x} - \mathbf{a})^\alpha$. Then
+
+$$
+    \partial^\beta T(\mathbf{x}) = \sum_{|\alpha| \leq n,\; \alpha \geq \beta} c_\alpha \frac{\alpha!}{(\alpha - \beta)!} (\mathbf{x} - \mathbf{a})^{\alpha - \beta}.
+$$
+
+Choose $\mathbf{x} = \mathbf{a}$: $\partial^\beta T(\mathbf{a}) = c_\beta \beta! = \partial^\beta f(\mathbf{a}) \implies c_\beta = \frac{\partial^\beta f(\mathbf{a})}{\beta!}$. Therefore we obtain
+
+$$
+    T(\mathbf{x}) = \sum_{|\alpha| \leq n} \frac{\partial^\alpha f(\mathbf{a})}{\alpha!} (\mathbf{x} - \mathbf{a})^\alpha. 
+$$
+
+*Theorem*: suppose $x \in D$ and the line segment $[\mathbf{a},\mathbf{x}]$ lies completely in $D$. Set $\mathbf{h} = \mathbf{x} - \mathbf{a}$. Then there is a $\theta \in (0,1)$ such that
+
+$$
+    f(\mathbf{x}) = T(\mathbf{x}) + \frac{1}{(n+1)!} \partial_\mathbf{h}^{n+1} f(\mathbf{a} + \theta \mathbf{h}).
+$$
+
+*Proof*: Apply Taylor’s theorem in 1D and the chain rule to the function $\phi : [0, 1] \to \mathbb{R}$ given by
+
+$$
+    \phi(\theta) := f(\mathbf{a} + \theta \mathbf{h}).
+$$
+
+## Other methods
+
+Creating multivariable Taylor polynomials by using 1D Taylor polynomials of the different variables and composing them. 
+
+### Example