Multivariable Functions

Topics:

• Be able to describe and sketch the domain of a function of two or more variables.
• Know how to evaluate a function of two or more variables.
• Be able to compute and sketch level curves & surfaces.

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We’ve already see some multivariable functions without putting them in that context. For example: !300 $z=9-x^2-y^2\text{ becomes...}$ $z=f(x,y)=9-x^2-y^2$

A function $f$ of two variables is a rule that assigns a unique value $f(x,y)$ to some set $D$ in ${\mathbb{R}}^2$. We call $D$ the domain of $f$, and the range of $d$ is the set of outputs. We often write:

$$z=f(x,y)$$

Similarly, in ${\mathbb{R}}^3$, A function $f$ of three variables is a rule that assigns a unique value $f(x,y,z)$ to each point in some set $D$. Here, we often write:

$$w=f(x,y,z)$$

The graph of a function $f$ of two variables with domain $D$ is the set of all points $(x,y,z)$ such that $z=f(x,y)$ and $(x,y)$ is in the domain of $f$.

Another option and perhaps the only way to visualize a multivariable function is to consider points that correspond to the same output value. If we join points with the same output value (z → the same elevation) If we join points with the same output value we get level or contour curves.

talk. about contour plot