What is the difference between a trace and a contour in calculus?

Question

As far as I can tell they're exactly the same thing, but the notes here discuss them as if they are separate:

The final topic in this section is that of traces.  In some ways these are similar to contours.  As noted above we can think of contours as the intersection of the surface given by $z=f(x,y)$ and the plane $z=k$.  Traces of surfaces are curves that represent the intersection of the surface and the plane given by $x=a$ or $y=b$.

Is the only difference whether we're holding an "input" to the function constant as opposed to the "output"? Generally the functions are defined by equations where any of the variables could be considered a function of the other two, so the distinction seems arbitrary? If there isn't a difference in denotation is there one of connotation?

Answer 1

Yes, traces are similar to contours, but contours can be a union of closed curves, while traces are never closed curves. In fact, traces are just graphs of single variable functions. For example, if $x=a$, then the trace is the graph of $z=f(a,y)$ where $z$ is a function of $y$ alone, and the definition of a function does not allow two values of $z$ for one value of $y$. Another example, $z=x^2+y^2$ has contours that are concentric circles. Here, for fixed values of $z,y$ if $x$ satisfies the equation, then so does $-x$. The key difference is that $z=f(x,y)$ where $f$ is a function as opposed to an example such as $r^2=x^2+y^2+z^2$ which only has contours since $z$ is not a single-valued function of $x,y$.

Answer 2

"but I don't know what those definitions are so I can compare" You just stated the definitions! A "contour" is the intersection of z= f(x, y) with z= constant. A "trace" is the intersection of z= f(x, y) with x= constant or y= constant. Those are the definitions.

The important difference comes from the definition of "function". The fact that z= f(x, y) means that one point (x, y) cannot correspond to two different values of z.

For example, let $z= x^2+ y^2$. The contour at z= 4 is the circle $x^2+ y^2= 4$. In general the contour at z= k is the circle $x^2+ y^2= k$. Both of those are closed curves. The trace at x= 1 is the parabola $z= 1+ y^2$ and the trace at y= 1 is the parabola $z= x^2+ 1$. Those are not closed curves.