AcademySeveral-Variable Calculus
Academy
Multivariable Functions
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
A multivariable function has more than one input. Its graph and level sets describe how outputs depend on several directions of change.
Notation
\(f(x,y)\)
function of two variables
\(D\)
domain in the input plane
\(z=f(x,y)\)
surface in three-dimensional space
The Core Method
Identify the allowed input pairs and evaluate the output. For two inputs, the graph is a surface.
Two-variable function
\[f:D\subseteq\mathbb R^2\to\mathbb R\]
Level curves are found by setting \(f(x,y)=c\), where \(c\) is a constant.
Worked Cases
Question
Find the domain of \(f(x,y)=\sqrt{9-x^2-y^2}\).
Answer
The expression under the square root must be non-negative: \(9-x^2-y^2\ge0\). Rearranging gives \(x^2+y^2\le9\). The domain is the closed disk of radius \(3\) centred at the origin.
Examples
Question
Describe the level curve \(f(x,y)=4\) for \(f(x,y)=x^2+y^2\).
Answer
Set \(x^2+y^2=4\). This is a circle of radius \(2\) centred at the origin.