# Exploring the Elegance and Potential of Discrete Calculus

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## Chapter 1: Understanding Discrete Calculus

The journey into the world of calculus can be fascinating, particularly when we explore its more generalized aspects. The beauty of discrete calculus lies in its potential to reshape our understanding of fundamental concepts.

Recently, while researching the Planck length—a theoretical minimum scale in our universe—I pondered whether the limitations of our modern gravitational theories stem from the continuous nature of traditional calculus. In calculus, we often allow variables to approach zero, including differences. But what if this approach doesn’t hold true for physical measurements?

To explore this, I began developing a version of calculus that operates solely with discrete intervals, rather than relying on limits. My intention was to apply this framework to physics using Planck units, potentially shedding light on what quantum gravity might entail.

Before diving into that discussion, I aim to present the mathematical theory underpinning discrete calculus, which stands as a captivating subject in its own right. Interestingly, my initial foray into this topic revealed some challenging (and possibly unsolvable) discrete differential equations when I attempted to address Einstein's field equations discretely. However, that narrative will unfold in a later piece.

It's important to note that discrete calculus, or the calculus of finite differences, is a recognized field, even if I was unaware of it while formulating my theory. After conducting thorough research, I found several established concepts, yet I also identified gaps in the literature, such as the absence of a generalized chain rule.

This doesn’t diminish the value of re-exploring this theory; rather, it highlights the enjoyment of engaging with it. In this article, we will derive a chain rule for discrete calculus and uncover many familiar principles from traditional calculus.

Studying this field can deepen our comprehension of calculus as a whole, as it represents a broader theoretical framework. You'll see that traditional calculus is merely a specific instance of this theory, with formulas applicable to countless variations of calculus, only one of which is the continuous form with which you are familiar.

### Section 1.1: Basics of Notation in Discrete Calculus

In mathematics, the choice of notation is crucial. It can either enhance understanding or lead to confusion. Therefore, we must be cautious not to conflict with the established notations in existing theories.

The theory of discrete calculus is built around a parameter I chose to denote as 'h,' which is common in the field. The value of 'h' significantly influences the outcomes of discrete derivatives and integrals.

Recall that in standard calculus, the differential operator is typically expressed as d/dx or D, with the resulting function denoted as f'(x). In our case, the discrete derivative will incorporate the parameter 'h' into its definition.

We define the discrete derivative with parameter h as follows:

This formula represents the slope of the secant line connecting the points (x, f(x)) and (x+h, f(x+h)). Notably, when h = 0, we revert to the conventional definition.

Most authors in the field restrict their analysis to h = 1, where the discrete derivative is referred to as the forward difference operator, denoted by Δ. In contrast, we will explore a more generalized approach, encompassing traditional calculus as well.

Let’s illustrate this with a familiar example: the exponential function. The results may surprise you, indicating that this broader context does not necessarily simplify matters! However, if we take the limit as h approaches 0, we arrive at the expected outcome, leaving it as an exercise for the reader.

Now that we've established our definition, it’s time to define the discrete integral using parameter h.

The first video provides an engaging exploration of discrete calculus concepts, making it an excellent starting point.

Continuing, as h approaches 0, the discrete versions of differentiation and integration converge to the familiar forms from traditional calculus. Furthermore, the summation in the definition of the discrete integral operates over integers ranging from 0 to (b-a)/h - 1, assuming (b-a)/h is an integer.

We define a discrete antiderivative as I f(x) = F(x, h), ensuring consistency with the broader theory, as evidenced by the fundamental theorem of discrete calculus.

### Section 1.2: Fundamental Theorem of Discrete Calculus

Let’s prove our initial theorem before presenting another example of a discrete derivative. The proof is straightforward, relying on our previously established definitions.

Using the discrete derivative definition, we derive:

Now, let's examine the application of the fundamental theorem. Consider f(x) = x². By employing the definition of the discrete derivative, we find that f*(x, h) = 2x + h.

Next, we aim to compute the discrete integral with parameter h of 2x + h from a to b, which, according to the fundamental theorem, should yield b² - a². Assuming (b-a)/h is an integer, the result aligns perfectly.

## Chapter 2: Rules of Discrete Calculus

We will outline the rules of discrete calculus, drawing parallels to the established rules in traditional calculus. These rules should revert to their familiar forms when we set h to 0.

Before proceeding, we will introduce two additional notation elements for clarity. It's important not to confuse the bar atop f with complex conjugation, for instance.

For linear functions, discrete differentiation aligns with ordinary differentiation—constant functions map to zero, and functions of the form ax map to a.

**Linearity**

As mentioned earlier, the first two rules, known collectively as linearity, are foundational in both calculus disciplines. The operators for discrete differentiation and integration are linear, meaning:

Next, we consider the product rule, which, while analogous to its familiar counterpart, presents unique characteristics.

**The Product Rule**

In keeping with traditional calculus, we have:

While this may appear unusual, the left side is symmetric in f and g, and the right side can be adjusted to maintain that symmetry. The proof is straightforward; we expand the right side using the definition and eliminate terms accordingly.

Let’s see this in action with the function h(x) = x³ = x²⋅ x, where we can verify that the product rule holds true.

**The Chain Rule**

One of the most beautiful results in this theory is the chain rule, which generalizes to the familiar form when h = 0:

To prove this, we define the right-hand side as y(x,h) and once again utilize the definition.

Now that we have established the chain rule, we can use it in conjunction with the product rule to derive the quotient rule for discrete calculus.

**The Quotient Rule**

With the chain rule in our arsenal, we can proceed to prove the quotient rule for discrete differentiation.

Let’s begin with a result easily derived from the definition, which serves as an outer function for discretely differentiating a function of the form 1/g(x). By employing the chain rule and the preceding result, we find that:

Now we can apply the product rule to the functions f(x) and 1/g(x) to derive the quotient rule.

This formulation returns us to the familiar quotient rule for differentiation when h = 0, allowing us to discretely differentiate various functions using our newly acquired tools.

The second video delves into the rules governing discrete derivatives, providing valuable insights for learners.

**Discrete Integration by Parts**

Having established a product rule for discrete differentiation, we can extend this concept to derive a rule for discrete integration by parts.

If we apply the product rule, selecting f to be integrated on both sides and then rearranging, we arrive at a compelling theorem.

This theorem can be expressed in terms of indefinite discrete integrals, and it retains its validity even with specified limits of integration.

Let’s apply this to compute the discrete integral of x⋅e^x. To utilize the theorem, we need to identify a discrete antiderivative of e^x, which is straightforward given our previous work.

Setting f*(x,h) = e^x and g(x) = x, we find that through discrete integration by parts, we achieve:

Here, c represents an arbitrary constant of discrete integration. For those skeptical (as I once was), applying the discrete product rule to the last result will reveal the cancellation of all troublesome h's, resulting in xe^x.

**Connecting to Traditional Calculus**

When we view discrete differentiation as a linear operator, it can be expressed in terms of the standard differential operator. While I won't delve deeply into this due to the article's length, this insight opens exciting avenues for exploration.

In particular, we can express:

Here, D denotes the conventional differential operator. By expanding this expression using the Maclaurin series for the exponential function, we observe:

This reinforces our previous findings, illustrating how these concepts interconnect beautifully.

**Looking Ahead**

This article merely scratches the surface of the intriguing world of discrete calculus. We have derived numerous rules that we can now build upon. The next step is to apply these findings to uncover new truths within this mathematical domain.

Potential areas of exploration include discrete differential equations, which could serve as the foundation for a fascinating article. Additionally, we might tackle Einstein's field equations using the Planck length as our smallest unit of distance, solving the corresponding system of discrete differential equations to uncover the metric tensor.

We could also pursue the development of generalized formulas, such as the discrete Taylor series, and investigate the feasibility of substitution within a discrete integral.

Feel free to share your thoughts in the comments about which direction you would like me to pursue for the next article on this topic.

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