I have been seeing this symbol ever since I started university and I am finding it hard to Google-fu what it is. Can someone tell me the name of it and hopefully the function of it as well?
It is the long vertical line at the end of the fraction before the subscript. I don't care for the equation itself (google images) but in my case I have something like dQ/dV|r and I don't know what i'm supposed to do with the r.
Thank You
$$A(S)\mid_{S=x}\,=A(x){}{}{}$$
The line denotes the value of the variable you have to put in the equation
In mathematics, this usage of a vertical bar means "restricted to". Example: Given a function $f$ mapping the set $\mathbb{R}$ of real numbers (its domain) into some set $S$, the expression $f\mid_{[0,1]}$ means a function whose domain is just the interval $[0,1]$ (a subset of $\mathbb{R}$) and whose values there agree with those of $f$.
As a specialization, if a single element $x$ of $f$'s domain is given as the restriction argument, then $f\mid_x$ may be taken to mean $f(x)$, and $f\mid_{x=x_0}$ is then the same as $f(x_0)$. A slight generalization of this is the shorthand notation used with definite integrals $$\int_a^b f(x) \mathrm{d}x = F(x)\mid_{x=a}^b = F(b)-F(a)$$ If $f$ has several parameters, say $x\in\mathbb{R}$ and $y\in\mathbb{R}$, then that $f$ has domain $\mathbb{R}\times\mathbb{R}$, and $f\mid_{x=x_0}$ is $f$ restricted to the domain $\{x_0\}\times\mathbb{R}$, whereas $f\mid_{y=y_0}$ is $f$ restricted to the domain $\mathbb{R}\times\{y_0\}$. You will notice the formal inconsistency with the single-parameter case, where the result of an explicit restriction to one point is usually not considered a function anymore, but just the value it takes there. As a workaround, you might write $f\mid_{\{x_0\}}$ to make sure the result is still a function.
In physics and engineering where symbols tend to represent physical entities rather than functions with a defined domain, there is an additional specialization: Restrictions of the form $\frac{\partial f}{\partial x}\!\mid_y$ where $x,y$ are variables (again usually representing physical entities) mean that $f$ shall be regarded as a function of $x$ and $y$ when doing the differentiation, with the consequence that the partial derivative $\frac{\partial f}{\partial x}$ is the same as the total derivative along a path where $y$ is held constant.
For example, in thermodynamics, the heat capacity (i.e. energy required per temperature change) for constant volume $V$ is typically written as $$C_V=\left.\frac{\partial U}{\partial T}\middle|_V\right.$$ meaning that the internal energy $U$ must be formulated as a function of temperature $T$ and volume $V$ before partial derivation can formally take place. This matters because it is in some sense more appropriate to regard the internal energy $U$ as a function $U(S,V)$ where $S$ is entropy, rather than as $U(T,V)$. That is, when given $U$ in terms of $S$ and $V$, you need to express $S$ in terms of $T$ and $V$ before you can calculate $C_V$.
