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

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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.