\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S05


Multivariable Functions

At the end of the course, each student should be able to…

  • S05: MulivarFunc. Sketch and analyze the domain, level curves, and graph of a two-variable real-valued function.

S05: Multivariable Functions

  • Functions of multiple variables are evaluated in the same way as single-variable functions.
    • Take \(f(x,y,z)=x^2yz-\frac{ze^x}{y^2+4}\) for example.
    • Then \(f(0,1,-3)=(0)^2(1)(-3)-\frac{(-3)e^{(0)}}{(1)^2+4}=-\frac{3}{5}\).
    • The domain of a function is the set of allowable inputs for that function.
  • Two-variable functions may be analyzed/sketched in 2D or 3D as follows.
    • The level curves of a two-variable function are the curves where \(f(x,y)=k\) for some value of \(k\). These curves lay in the domain and illustrate a topographical map of the function’s values.
    • The graph of a two-variable function is the surface \(z=f(x,y)\). It may be sketched from the equation \(z=f(x,y)\) or by lifting the level curves to appropriate heights in \(xyz\) space.

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 13.1 (exercises 1-16, 31-52)