Partial derivative "evaluated at" notation vs. function parameter notation

Question

In a text by a colleague, I have seen

$$\left. \frac{\partial f}{\partial s} \right|_{(a, b)}.$$

Is this notation equivalent to using:

$$\frac{\partial f}{\partial s} (a, b)?$$

Answer 1

Yes, they may be understood as the same thing; they are both the evaluation of the partial derivative of $f$ respective to $s$ evaluated at $a,b$ where the order of parameters of $f$ to be replaced is implicit from context.   That is, when $f$ is established as shorthand for $f(s,t)$ or such.

$$\left.\dfrac{\partial f}{\partial s}\right\vert_{(a,b)} ~=~ \dfrac{\partial f}{\partial s}(a,b) ~=~ \left.\dfrac{\partial f(s,t)}{\partial s}\right\vert_{s=a, t=b} $$