"Evaluated at" or "at" notation Normally a variable that is a function another variable would be represented as in the following fashion: $ V(t) $ (voltage as a function of time). However, my engineering professor (who also wrote the textbook) likes to use the vertical bar notation instead, so the flux at $ r = R_0 $ is $ N|_{r=R_0} $ instead of $ N(R_0) $ or $ N(r=R_0) $. Likewise, $ C|_z - C|_{z+\Delta z} $ and so on.
Personally, I would rather reserve the vertical bar for a more complex expression like evaluation of a derivative:
$$
\left.\frac{df}{dt} \right|_{t=4} = \left. t^2+3t\right|_{t=4}
$$
Which is the preferred notation? If it's just a matter of style, then what are the pros and cons of each option?

I've usually seen that notation mean "restricted to", which isn't too far from "evaluated at" in meaning, but it is more general. For instance,
$$
\left. f\right|_{[0,1]}
$$
could be interpreted to be a function with values that agrees with $f$'s values, but is only defined on the interval $[0,1]$.
One advantage is that you're not dependant on a function name in order to write down both the function and the evaluation point, as is witnessed by
$$
\left. t^2 +3t\right|_{t=4}
$$
Which with normal notation would be cumbersome to separate from, let's say
$$
\left. t^2 +12\right|_{t=4}
$$
One disadvantage is that you'd need to add brackets to really be able to tell where the expression begins.
Another one is clarity: it's not as much in use, and you would therefore have to explain it every time you use it. Composition of functions might also get messy if you get right down to it, with nested bars and subscripts. Compare $f(g(x))$ to
$$
f|_{g|_x}
$$
