Functions of Several Variables

The central idea

A scalar field $f:D\subset\mathbb R^n\to\mathbb R$ assigns one number to each input vector; a vector-valued map may instead output $\mathbb R^m$. For $f(x,y)$, the graph lies in $\mathbb R^3$, while a level curve $f(x,y)=c$ lies in the input plane. Domain restrictions apply jointly to all coordinates.

Definitions, hypotheses, and notation

Slices and level sets answer different questions. A trace such as $x=a$ cuts the graph by a vertical plane and produces an output curve. A level set cuts by a horizontal plane $z=c$ and projects to points of equal output. Confusing them can reverse which variable is fixed.

For $f:\mathbb R^3\to\mathbb R$, the graph would require four dimensions, so level surfaces in ordinary three-dimensional input space become the primary visualization. Calculus definitions should therefore be expressed without depending on a drawable graph.

Conceptual meaning

Multiple inputs describe systems with several independent controls, such as temperature over a map or height over a plane. Graphs add an output axis; contour plots encode the same output by labelled level sets without leaving the input space.

A dependable method and decision rule

1. State input and output dimensions.
2. Find domain inequalities from radicals, denominators, and logarithms.
3. Analyze traces by fixing one coordinate.
4. Analyze level sets by fixing the output.
5. Use symmetry and boundary behavior to connect formula, graph, and contour map.

Fully worked example

Common mistakes and why they fail