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 […]