Functions of Several Variables

Multivariable functions.

Graphs of 2-var functions are surfaces in 3D space

• Ex: $z=f(x,y)=2x^2+2y^2-4$

Equation of a plane: $ax+by+cz$

• Solving for $z$ allows us to use function notation $f(x,y)=Ax+By+D$ $f(x,y,z)$ would be a 4D surface!

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Level Curves

Level curves of a 3D function $z=f(x,y)$ are 2D curves we get by setting $z=k$, where $k$ is any number. Thus, equations of level curves are $f(x,y)=k$.

The intuition for this like slicing a 3D object at a certain height.

• Contour/level represents the intersection given by surface $z=f(x,y)$ and $z=k$ For higher dimensions like $f(x,y,z)$, it would be a level surface instead of a level curve

A plot of various level curves is called a contour plot.

Example: Cone Level Curves

Identify the level curves of $f(x,y)=\sqrt{x^2+y^2}$. The function is then: $z=\sqrt{x^2+y^2}$ Using $k$ instead: $k=\sqrt{x^2+y^2}$ Simplifying: $x^2+y^2=k^2$

Traces:

• Curves that represent the intersection of the surface given $z=f(x,y)$ and places given by $x=a$ or $y=b$

Trace Example

Let $f(x,y)=10-4x^2-y^2$ for $x=1$ and $y=2$. $x=1⟶z=f(1,y)=10-4(1{)}^2-y^2=6-y^2$ $y=2⟶z=f(x,2)=10-4x^2-2=6-4x^2$