12. Multivariable functions

Today, we will see a new type of function. A multivariable function is a function with more than one independent variable (input) and a single dependent variable (output). Our notation is $\large z = f(x,y)$. There are examples all around us:

The temperature depends on location and time, so $\large T=f(x,y,z,t)$.
The price of my auto insurance depends on my age ($\large a$), my driving record ($\large r$), and my gender ($\large g$), so this function is $\large P=f(a,r,g)$.
The probability of a random person being over $\large x$ inches tall and weighing over $\large y$ pounds obviously depends on both $\large x$ and $\large y$: $\large P = f(x,y)$.

The ideas of domain and range carry over naturally to multivariable functions. For example, the domain of $\large f(x,y)=\sqrt{9-x^2-y^2}$ is $\large \lbrace(x,y):x^2+y^2\le 9\rbrace$ (the set of all points that lie within or on the circle of radius 3).

$\large z=f(x,y)$ represents a surface

We've seen planes, cylinders, and quadric surfaces. There are infinitely many more surfaces out there! In the function $\large z=f(x,y)$, every $\large (x,y)$ point gives rise to a $\large z$ value. The resulting plot is a surface. Here are two examples:

$\large f(x,y)=7xye^{-(x^2+y^2)}$ (you may recognize this one from the main image in CalcPlot3D):

surface1

$\large f(x,y)=x\cos(y)$:

surface2

We typically just have a computer plot these for us, but the idea of traces is still helpful in understanding the shapes of surfaces. Let's look at The second surface above, $\large z=x\cos(y)$. If we let $\large x=3$, we get $\large z=3\cos(y)$. We can see that trace on the left below. If we let $\large y=2$, we get $\large z=\cos(2) x$, a line through $\large z=0$. We can see that trace on the right below.

surfaceTrace1 surfaceTrace2

Contours: another approach to visualizing $\large z=f(x,y)$

The horizontal traces play a special role with $\large f(x,y)$. The intersection of the surface $\large z=f(x,y)$ and the plane $\large z=c$ is called a contour. Here's a bunch of contours for the function $\large f(x,y)=\sin(x)+\cos(y)$:

contour1

The projection (shadow) of a contour onto the $\large xy$-plane is called a level curve. A plot in the $\large xy$-plane of level curves for equally spaced values of the constant $\large c$ is called a contour map. Here's the same contour map for the contours shown above:

contour2

Each line is a line of constant elevation (constant $\large z$). Hikers called these topographical maps. Here's a topo map of Golden, CO:

goldentopo

If you walk along a single contour, you do not change elevation. A surface is steep where contours are close together.

Finally, a video showing how to create a contour map for a surface (we can do a few contours this way, but, really, we want a computer to create these for us):