1 KiB
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,fis a scalar function, - for
n>1,fis 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.