Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

Question

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is exactly expanded and what its definition is. What if we had equations like follows (even if they do not necessarily make sense):

1. $\left.\dfrac{d}{dt}\right|_{a=0}f(a+tv)$

2. $\left.\dfrac{d}{dt}\right|_{s=0}f(a+tv)$

3. $\left.\dfrac{d}{dt}\right|_{t=0}f(a+v)$

4. $\left.\dfrac{d}{dt}\right|_{t=0}f(a)$

What are they equivalent to by definition after expansion?

To clarify myself, is the general definition $\left.\dfrac{d}{dt}\right|_{s=u}f(a+rv)=\lim \limits_{s\to u}\dfrac{f(a+rv)-f(a)}{t}$?

Answer 1

1. means that you derive $f(a+tv)$ w.r.t. $t$ (this gives $Df(a+tv)[v]=D_vf(a+tv))$) and then you evaluate the latter at $a=0$, which gives $Df(tv)[v]=D_vf(tv)$.
2. means that you derive $f(a+tv)$ w.r.t. $t$ (this gives $D_vf(a+tv)$) and then you evaluate the latter at $s=0$, which gives $D_vf(a+tv)$ since the previous results doesn't depend on $s$.
3. means that you derive $f(a+v)$ w.r.t. $t$ (this gives $0$) and then you evaluate the latter at $t=0$, which gives $0$ again.
4. means that you derive $f(a)$ w.r.t. $t$ (this gives $0$) and then you evaluate the latter at $t=0$, which gives $0$ again.

So the bar notation represents the evaluation at the specified value after the differentiation has been applied. Explicitly:

$\left.\dfrac{d}{dt}\right|_{s=u}f(a+rv)= \left.(\dfrac{d}{dt} f(a+rv))\right|_{s=u}$

Answer 2

By definition, 1. is the derivative of $f(tv)$, i.e, $vf^\prime (tv)$. For 2., if $s\neq t$, then the result is $0$. Assuming $v\neq v(t)$ gives $3.$ as $0$, and $4.$ is simply $0$ (it is obvious).

Answer 3

For any function $f$ of $t$, the derivative $\frac{d}{dt} f$ is evaluated at a point. Bar notation simply indicates: take the derivative at this specified value of t. Thus the expansion is simply the definition of the derivative of the given function, with respect tot eh given variable, at the specified value of that variable.

Answer 4

First equation:

\begin{equation} \left.\dfrac{d}{dt}\right|_{a=0}f(a+tv) = \frac{d}{dt}f(tv)=vf'(tv) \end{equation}

where $f'(x)\equiv\frac{d}{dx}f(x)$.

Second equation:

\begin{equation} \left.\dfrac{d}{dt}\right|_{s=0}f(a+tv) = vf'(a+tv)|_{s=0} \end{equation} whatever $s$ may be.

Third equation: \begin{equation} \left.\dfrac{d}{dt}\right|_{t=0}f(a+v)=0 \end{equation} unless $a$ or $v$ depends on $t$.

Finally, fourth equation: \begin{equation} \left.\dfrac{d}{dt}\right|_{t=0}f(a)=0 \end{equation} unless $a$ depends on $t$.

Also, your definition is a special case of second equation with $s=t$: \begin{equation} \left.\dfrac{d}{dt}\right|_{t=0}f(a+tv) = vf'(a) \end{equation}