## 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 […]