### 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 $\phi\in[0,2\pi)$;
And there are more, involving spherical coordinates and vectors:
• $r=1$ (where $r$ is the spherical radial coordinate);
• $\rr = x\,\xhat + y\,\yhat + z\,\zhat$, with $x$, $y$, $z$ given as above;
All of these descriptions describe the same surface. Which representation is best for a given problem depends on the circumstances.

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 $$ax+by = e \label{lineab}$$ for some constants $a,b,e$, any plane can be written as $$ax+by+cz = e \label{planeabc}$$ for some constants $a,b,c,e$. For (nonvertical, i.e. $b\ne0$) lines, one often writes (\ref{lineab}) in slope-intercept form as $$y = Ax + C$$ with $A=-a/b$ and $C=e/b$. Similarly, nonvertical planes ($c\ne0$) are often written in the form $$z = Ax + By + C$$ 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.