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
,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
.