## Finite Differences

Here are a few notes about the symbol $ \Delta$. This symbol appears frequently in physics. Let $ \alpha$ be a real number. This statement is equivalent to $ \alpha \in \mathbb{R}$. Then $ \Delta \alpha$ can be defined as follows.

**Definition 1** Define a finite difference as $ \Delta \alpha \equiv \alpha_2 – \alpha_1$. Here, $ \alpha_1 \in \alpha$ and $ \alpha_2 \in \alpha$.

Something to notice is that if $ \alpha_1 = 0$, $ \Delta \alpha = \alpha_2$. This is related to the content presented in Derivation #1. For example, in that post there are equations written in terms of $ X$ which are used to write equations in terms of $ \Delta X$. Using the notation in Definition 1, this can be done because $ X_1$ can be set to 0 as the light is at the origin at $ t=0$. If $ X_2$ is set to $ X$, then $ \Delta X = X$ in this case. If, instead, $ X_2$ were set to 0 and $ X_1$ were set to $ X$, then one would have $ \Delta X = -X$. The first case was used in Derivation #1. It is interesting to note that there is not a functional dependence of $ X$ on time, but there appears to be a subtle, implied association between position and time.

Something must be said about the independent variable with which the corresponding function changes. Suppose one has a function $ \beta = \beta(t)$ in which $ t \in \mathbb{R}$. Then I can define the following constraint on the independent variable. Using Definition 1, it is clear that $ \Delta t = t_2 – t_1$. Here is the constraint used in Calculus.

**Constraint 1** $ t_2 > t_1$.

Modern Calculus does *not* utilize the following, opposite constraint on the independent variable.

**Constraint 2** $ t_1 < t_2$.

To better understand this, consider an example of two points plotted on a coordinate system. Here is a picture of two different cases.

Both cases depict the same points, but the labelling scheme is different.

These two points are $ (t_1, \beta_1(t_1) )$ and $ (t_2, \beta_2(t_2) )$. It is important to remember that a function such as $ \beta(t)$ correlates input points with output points so it does not make sense to write, for instance, $ (t_1, \beta_2(t_2) )$.

In the first case (blue), $ t_2 > t_1$ so Constraint 1 is met. As drawn, $ \beta(t_2) > \beta(t_1)$. Therefore, using Definition 1 the ratio $ \frac{\Delta \beta(t)}{\Delta t} \equiv \frac{\beta(t_2) – \beta(t_1)}{t_2 – t_1} >0$.

In the second case (orange), $ t_1 < t_2$ so Constraint 2 is met. As drawn, $ \beta(t_2) < \beta(t_1)$. Therefore, using Definition 1 the ratio $ \frac{\Delta \beta(t)}{\Delta t} \equiv \frac{\beta(t_2) – \beta(t_1)}{t_2 – t_1} > 0$.

These two ratios for $ \frac{\Delta \beta(t)}{\Delta t}$ are the same regardless of whether the labelling convention of $ t_2 > t_1$ or $ t_1 > t_2$ is chosen. This is due to the input-output nature of functions; if the independent variables are labelled differently, the dependent variables are labelled differently as well.

This means that all the information about how a function changes with respect to a certain variable can be obtained by using either Constraint 1 or Constraint 2. Both constraints do not need to be considered for a given set of points comprising a function.