When can we modify $a \propto \ b$ adding $\delta $?

Question

This is a general, theoretical question about formalization of concepts, it is difficult for me to explain it adequately, please, if I fail, tell me in a comment what is not clear or feel free to edit it.

Maths can be applied to different disciplines, can you explain how to correctly formalize an increase (change) of a quantity $b$ when that change is not a rate of change depending on time?

• can you, for starters, say which one would be more appropriate here: $\delta$ or $\Delta$, or none of the two?

Consider please an example in real life, or in Economics, and suppose we have a relation of proportionality $a \propto (\delta) b$ (a relation in which time is not relevant) for example:

each Basket produced is paid Wages 5 euro: 1 B = 5 [= k] €, k = W/ B, $\Rightarrow W = kB$,

In this case time is not at all relevant, you can produce a B in 5 minutes or 1 hour or 1 day and the reward does not change. On the other hand, the wages can be paid instantly with a cheque, in a second or two with a roll of banknotes or the notes can be slowly counted and handed over one by one and delivered in ten seconds, or delivered in the course of one day or a week.

If we represent this relation with the proportional symbol, does the symbol $\delta$ make any difference? and which symbol is appropriate $\delta$ or $\Delta$?

Does this comment exclude the appropriateness of $\delta$, since my example obviously does non refer to the *traditional differential in calculus"?

See here for the uses of δ and Δ in mathematics; I assume that you are meaning the "traditional" differential of calculus : dx,dt,dv,…. – Mauro ALLEGRANZA

Should we replace, then $\delta/$ with $ \Delta B$ , or drop them altogether?

W [ages] are proportional to B [askets] 1 B = 5€ : $\Rightarrow W \propto (\delta) B$, this means that there are already a number of baskets (10) in the workshop $B_i$ and, if the basket become 12 the wages will be $[k=]5* (B_f-B_i: 12-10=)2 =10$ euro

But, if there are 10 baskets, that means that 50 euro have been already paid, so

• 1) can we arbitrarily change $\Rightarrow W \propto \delta B$ to $\Rightarrow \delta W \propto \delta B$?, or
• 2) can we arbitrarily change $\Rightarrow W \propto \delta B$ to $\Rightarrow W \propto B$?

One final question:

• 3) Can we apply your answers, conclusion to other cases/ quantities: if, instead of B[basket], we have v[elocity] can we consider $\delta v$ just $v_f-v_i$, that is refer to 'change of velocity' as 'velocity' and not 'rate of change of v' = acceleration?, or in that case we can formalize it only as $v_f-v_i$?

Answer 1

I'm not sure if this answers the question but maybe it can shed some light on the problem.

It seems here that the main point of the question is how to handle varying quantities. In mathematics there is the central concept of function; loosely speaking, a function is a quantity $y$ that depends on another quantity $x$ which varies in a certain environment. The quantity $x$ is called the independent variable and $y$ the dependent variable. In mathematics we write $y=f(x)$ to express the fact that $y$ and $x$ are related via a causal bond.

The dependent variable's environment is a set, let's call it $D$. In many cases $D$ has a "good" structure, in the sense that it's possible to define operations such as the sum or difference between elements of $D$, or even measure the size of some subsets of $D$ (after having defined what is the "size" of a subset).

Think of the set $\mathbb{R}$ of the real numbers, for instance: given two points $x_1$ and $x_2$ in $D$, we can define an operation that takes the ordered couple $(x_1,x_2)$ and returns the difference $x_2-x_1$.

We call this quantity $\Delta x$: \begin{equation} \Delta x := x_2-x_1 \end{equation}

Note that with this symbology we understate that we are referring to two definite elements of the set $D$, taken in a particular order.

If we have a functional relation between $y$ and $x$, say $y=f(x)$, and also the environment of $y$ has a "good enough" structure, we can define $y_1=f(x_1)$ and $y_2=f(x_2)$, and also

\begin{equation} \Delta y := y_2-y_1 \end{equation}

Now we can define two concepts: the change or increment of the dependent variable or quantity $y$, and the rate of change of the same quantity.

The change of $y$ is simply $\Delta y$. This notation hides a lot of information, because since the "$\Delta$" notation implies choosing two elements of a set, we must realize that we are considering two points $y_1$ and $y_2$; these two points in turn depend on two values of the dependent variable $x$, namely $x_1$ and $x_2$.

So really, we must look at $\Delta y$ as \begin{equation} \Delta y = f(x_2)-f(x_1) \end{equation}

We are now lead to consider how $y$ varies with respect to the variation of $x$. In our framework we are dealing with numbers, and to implement this concept we may invoke the tool of division, as it is the appropriate formalization when dealing with relative measurements (as percentages are relative measures opposed to absolute values):

\begin{equation} \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} \end{equation}

This is exactly the rate of change of $y$ with respect to $x$. (Note that speaking of the rate of change of $x$ with respect to $x$ itself is somewhat vacuous, since the environment of $x$ is not varying "against some kind of different environment" as $y$ does. Conceptually it is the same as saying that the rate of change of $x$ with respect to $x$ is $1$)

The link between the change of $y$ and the rate of change of $y$ with respect to $x$ is ultimately the unit of measure: we can think of the change of $y$ as the unitary rate of change if we normalize to $1$ the difference $\Delta x$ by choosing appropriate units in the set $D$.

The operator $\delta$ is similar in concept to the $\Delta$ operator but it requires an additional property on the set $D$ and on the environment of $y$, namely these sets must have "a lot" of points, i.e. more than countably many. The $\delta$ operator is a kind of difference operator for infinitesimal increments. Maybe you can think of $\delta x$ as $\Delta x$ when $x_2$ and $x_1$ are infinitely close: such a situation arises (in a meaningful sense) when $D$ has a continuum of points.

Usually we denote $\delta x$ with $dx$; in this setting the ratio \begin{equation} \frac{dy}{dx} \end{equation} is called the istantaneous rate of change of $y$ with respect to $x$, and is actually the derivative of $y$ with respect to $x$.

Answer 2

I'll give you some advice, rather than a definitive answer, we must wait for a more qualified member:

If you use either symbol you carry a particular connotation, as it was hinted in a comment

you are meaning the "traditional" differential of calculus : dx,dt,dv,….

.

Should we replace, then $\delta/$ with $ \Delta B$ ?

W [ages] are proportional to B [askets] 1 B = 5€ : $\Rightarrow W > \propto (\delta) B$, this means that there are already a number of baskets (10) in the workshop $B_i$ and, if the basket become 12 the wages will be $[k=]5* (B_f-B_i: 12-10=)2 =10$ euro

Probably it is better to use another notation

$(W_f - W_i) \propto (B_f -B_i)$ or $(W_1 - W_0) \propto (B_1 -B_0)$

Answer 3

Using the formalism set up by marco trevi (which is very standard and well-written, by the way), I will answer the questions you asked in the OP.

I will make one further assumption; that when we write $W$ we mean $W$ collected at the same point as the final point of $\delta B$. This is because we will think of $D=\Bbb R$ for these functions as being "time", and in this case we are saying that we can only collect measurements "from the past", not "from the future".

1) Can we arbitrarily change $\Rightarrow W \propto \delta B$ to $\Rightarrow \delta W \propto \delta B$?,

No. We choose two times $x_1$ and $x_2$ at which to consider $W$ and $B$; in this case, the question is if $W(x_2)=k[B(x_2)-B(x_1)]$, is it true that $W(x_2)-W(x_1)=k[B(x_2)-B(x_1)]$?

This is not true unless $W(x_1)=0$; in other words, if initially there were no wages offered.

2) Can we arbitrarily change $\Rightarrow W \propto \delta B$ to $\Rightarrow W \propto B$?

No. The question, formalized, is if $W(x_2)=k[B(x_2)-B(x_1)]$, is it true that $W(x_2)=kB(x_2)$?

This is not true unless $B(x_1)=0$, in other words, if initially no baskets had been produced.

3) Can we apply your answers, conclusion to other cases/ quantities: if, instead of B[basket], we have v[elocity] can we consider $\delta v$ just $v_f-v_i$, that is refer to 'change of velocity' as 'velocity' and not 'rate of change of v' = acceleration?, or in that case we can formalize it only as $v_f-v_i$?

I'm not sure how to give a yes or no answer to the question as a whole, so:

• Yes, we can apply our answers to other quantities; you notice that very little in the above arguments had to do with wages or baskets.
• Yes, we can consider $\delta v = v_f-v_i$, or in the formalism we have used here, $v(x_2)-v(x_1)$.
• This quantity has units of velocity, but it is not really the same kind of velocity. There is some serious mathematics happening here which you may or may not be interested in. If not, it's good enough to know that you are describing a "change in velocity".
• No, $\delta v$ is not a rate of change of $v$, and therefore is not an acceleration. Note that if you did want a rate of change of $v$, that would be a relative measurement ($\delta v$ relative to the time duration considered). Therefore it would be $\frac{v(x_2)-v(x_1)}{x_2-x_1}$
• As I said above, it is possible to find another formalization for $\delta v$, but it probably would confuse the issue more than clarify it.