Functions of several variables

Definition: let D \subseteq \mathbb{R}^m with m>1, and f: D \to \mathbb{R}^n then f is a function of several variables where:

for n=1, f is a scalar function,
for n>1, f is a vector valued function.

Definition: the domain convention specifies that the domain of a function of m variables is the largest set of points for which the function makes sense as a real number, unless that domain is explicitly stated to be a smaller set.

Graphical representations of scalar valued functions

Graphs

Definition: let D \subseteq \mathbb{R}^2 and let f: D \to \mathbb{R} then G_f := \big\{\big(x, y, f(x,y)\big) \;\big|\; (x, y) \in D\big\} is the graph of f. Observe that G_f \subseteq \mathbb{R}^3.

Level sets

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}).

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