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

\[ \partial^\alpha f(\mathbf{x}) = \partial^{\alpha_1}_{x_1} \dotsc \partial^{\alpha_n}_{x_n} f(\mathbf{x}). \]