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Math101learn.math101.caFunctions of Several Variables
A rigorous, example-driven guide to functions of several variables, including hypotheses, method choice, verification, and practice.
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
- State input and output dimensions.
- Find domain inequalities from radicals, denominators, and logarithms.
- Analyze traces by fixing one coordinate.
- Analyze level sets by fixing the output.
- Use symmetry and boundary behavior to connect formula, graph, and contour map.
Fully worked example
Graphical or geometric meaning
Higher contour values shrink toward the origin for the hemisphere, while lower levels form larger circles near the domain boundary. Dense contour spacing indicates steep change in height.
Common mistakes and why they fail
Verification and reasonableness checks
- Substitute level-set points back into the function.
- Verify traces and levels agree at their intersections.
- Label every representation with its ambient coordinate space.
Domain and level sets reveal the geometry
A multivariable formula assigns one output to points in a domain. Determine that domain by combining all restrictions: denominators nonzero, even-root radicands nonnegative, and logarithm arguments positive. Its graph for $z=f(x,y)$ is a surface, while level curves $f(x,y)=k$ are horizontal slices projected into the plane. Evaluating several levels often communicates shape more clearly than plotting isolated points. Units also matter: changing one input while fixing others defines a partial response, not a total change along a path. Before applying calculus rules, mark boundary points where formulas or derivatives may fail. An algebraic simplification does not restore points excluded by the original expression, just as in one-variable calculus.
Practice
- Find the domain of $\ln(4-x^2-y^2)$.
- Describe levels of $x^2+y^2$.
- Where does a graph of $f:\mathbb R^2\to\mathbb R$ live?
Answers and brief solutions
- $x^2+y^2<4$.
- Circles centered at the origin for nonnegative levels.
- In $\mathbb R^3$.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the domain of f(x,y)=√(9−x²−y²)?
- Require 9−x²−y²≥0.
- Rearrange to x²+y²≤9.
- The domain is the closed disk of radius 3.
End of lesson
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