Im really stuck on this problem, its for an applied mathematical modelling class. I spoke to my professor about it and he said to really have a think about it, as if its a trick question or has a trivial answer.
the question asks to: show and explain how eq (1) is re expressed as eq (2)
Im thinking something along the lines of: $$dS=S_{t+Δt}-S_t$$ and $$dt=Δt$$ because dS represents a change in S ($S_{t+Δt}-S_t$ is a change in S) and dt represents a change in t (Δt is a change in t) so $$\frac{S_{t+Δt}-S_t}{dt} = F_{s,in} - F_{s,out}$$ $$\frac{S_{t+Δt}}{dt} = S_t + (F_{s,in} - F_{s,out})$$ $$S_{t+Δt} = S_t + (F_{s,in} - F_{s,out})*Δt$$
even if this arithmetic is correct, I still dont know how to explain it. I could just be over thinking it.
-Thanks in advance
You've got the point, we just need to remember a bit of theory from calculus 1, $$ \frac{dS}{dt} = \lim_{\Delta t \to 0} \frac{S(t + \Delta t) - S(t)}{ \Delta t} $$ Here we're using $S(t) = S_t$ and $S(t + \Delta t) = S_{t + \Delta t}$. So as $\Delta t \to 0$, the two formula are identical.
In theory, we can measure increments that are arbitrarily close in time ($ \Delta t \to 0$). In practice, this absolutely never happens and you pay \$\$\$ for each decimal place of precision. The underlying process (fluid flux) is continuous and smooth, so our limit should converge if we could only take a finer time step.
If you assume your phenomena is smooth, you could consider the Taylor series expansion and use a series representation to estimate the order of accuracy (big O notation) convergence of the approximation, but you've got the idea for a modeling class.