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### Surfaces

There are many ways to describe a surface. How many can you think of? Take a moment to make a list before reading further.

Consider the following descriptions of a surface:

- the upper half of the unit sphere;
- $x^2+y^2+z^2=1$ with $z\ge0$;
- the graph of $z=+\sqrt{1-x^2-y^2}$;
- $x=\sin\theta\cos\phi$, $y=\sin\theta\sin\phi$, $z=\cos\theta$, with $\theta\in[0,\frac{\pi}{2}]$ and $\phi\in[0,2\pi)$;

- $r=1$ (where $r$ is the
*spherical*radial coordinate); - $\rr = x\,\xhat + y\,\yhat + z\,\zhat$, with $x$, $y$, $z$ given as above;

The simplest surfaces are those given by holding one of the coordinates constant. Thus, the $xy$-plane is given by $z=0$. Just as any line (in the $xy$-plane) can be written in the form \begin{equation} ax+by = e \label{lineab} \end{equation} for some constants $a,b,e$, any plane can be written as \begin{equation} ax+by+cz = e \label{planeabc} \end{equation} for some constants $a,b,c,e$. For (nonvertical, i.e. $b\ne0$) lines, one often writes (\ref{lineab}) in slope-intercept form as \begin{equation} y = Ax + C \end{equation} with $A=-a/b$ and $C=e/b$. Similarly, nonvertical planes ($c\ne0$) are often written in the form \begin{equation} z = Ax + By + C \end{equation} where now $A=-a/c$, $B=-b/c$, and $C=e/c$. $A$ and $B$ determine the slopes of such planes in the $x$ and $y$ directions, respectively; two planes with the same values of $A$ and $B$ are parallel.

Many surfaces, including nonvertical planes, are the graph of some function, typically written $z=f(x,y)$. For example, $z=ax^2+by^2$ describes an *elliptic paraboloid*; the special case $a=b$ is often simply called a *paraboloid*.

But not all surfaces are expressible as the graph of a function, and even those which are often have simpler representations. For example, the upper half of a sphere was written above as the graph of $z=\sqrt{1-x^2-y^2}$, yet the equation on the previous line, obtained by squaring both sides, is simpler. And the sphere itself can not be written as the graph of a single function, yet the same equation can be used for the entire sphere, namely $x^2+y^2+z^2=1$. More generally, we can graph *equations*, not merely *functions*.

How can you visualize and graph such surfaces? One way is to consider what the surface looks like if one of the variables is held constant; the resulting curves are called *traces* of the surface. A better name might be *slices*, as this method amounts to slicing the surface parallel to some plane, then imagining how to stack the resulting slices.