Manipulation of an expression with derivatives: which sign is more suitable, "d" or "∂"? I am writing a scientific paper about Mechanics of Beams. At a given point, I discuss the quadratic strain in function of $u(x,t)$ and $w(x,t)$, which refer to the beam's motion in function of position $x$ and time $t$. The quadratic strain reads:
\begin{equation}
    \varepsilon_q = \dfrac{\left(du+dx\right)^2+dw^2-dx^2}{dx^2} \cong \dfrac{\partial{u}}{\partial{x}}+\dfrac{1}{2}\left(\dfrac{\partial{w}}{\partial{x}}\right)^2.
\end{equation}
As you can see, on the right side I use " $\partial$ ", which is suitable to the fact that $u(x,t)$ and $w(x,t)$ depend on both $x$ and $t$. However, on the left side I use " $d$ ", which is suitable to the fact that $du = u(x+dx)-u(x)$ and $dw = w(x+dx)-w(x)$. I wonder if it would not be more accurate to write:
\begin{equation}
    \varepsilon_q = \dfrac{\left({\partial}u+{\partial}x\right)^2+{\partial}w^2-{\partial}x^2}{{\partial}x^2} \cong \dfrac{\partial{u}}{\partial{x}}+\dfrac{1}{2}\left(\dfrac{\partial{w}}{\partial{x}}\right)^2,
\end{equation}
even though I have neever seen $\partial{x}$, $\partial{u}$ or $\partial{w}$ written aside - let us say out of a "derivative fraction" - as we commonly see in math or engineering texts for $dx$, $du$ and $dw$. For example, I have often seen the expression $du = u(x+dx)-u(x)$; but never $\partial{u} = u(x+\partial{x})-u(x)$.
That being said, which expression do you think I should choose?
 A: To address your concern directly, it would not be more accurate to replace $du$ with $\partial u$. The partial derivative notation $\partial u$ is meaningless without a corresponding $\partial x$ indicating which variable is meant to vary.
It can only be made meaningful if the function $u$ is interpreted as a one-variable function anyway. And that's precisely what's happening here - the time dependence of $u$ is being implicitly suppressed, and the $du$ refers to only the variable $x$.
Anyone who fails to understand the $du$ notation would certainly fail to understand the $\partial u$ notation.
A: The standard notation for partial derivatives is terrible, as the denominator contains information about the numerator, and, because of that, they can never be split.
A notation I have suggested is to subscript partial differentials with the variables that were allowed to change.  So, for $f(x, y)$:
$$ df = \partial_x f + \partial_y f $$
In other words, the total differential is the sum of all of the partials.
So, what is normally thought of as $\frac{\partial f}{\partial x}$ is actually more explicitly written as $\frac{\partial_x f}{dx}$.  This allows the differentials to be treated algebraically.
Using the notation above, you can be more clear as to whether the differentials being considered are partial or total, and, if partial, which one was allowed to vary.
