The Fundamental Theorem of Calculus

What exactly is the “Fundamental Theorem of Calculus,” and what justifies its name? Derivatives are defined using limits, and while it’s relatively straightforward to manipulate algebraically to find a derivative, integrating rigorously is a more challenging endeavor. The connection between derivatives and integrals is profound; this theorem reveals that they are inverse operations.

Calculating an integral by summing infinitesimal quantities is quite complex. Thus, rather than performing a Riemann summation, we typically compute the antiderivative. Since the result of an integral is equivalent to reversing a derivative, we opt to apply the process of undoing the derivative instead of summation.

This is why it holds such fundamental importance in calculus; we rarely compute an integral through summation but rather as the inverse of differentiation (by recalling the outcomes of derivatives). Without recognizing that derivatives and integrals are reciprocal, we would struggle to utilize antiderivatives in lieu of integrals, significantly complicating our typical approach to integration.

Criteria for Beauty in Mathematics

What can we define as beautiful in mathematics? Many individuals perceive math as merely about numbers and performing operations like addition and multiplication. While these skills are essential, the ultimate objective is to uncover broader truths about structures, allowing us to comprehend them in an abstract manner.

By establishing certain conditions, we aim to derive all possible implications. The goal isn’t to tackle each individual case in a tailored manner but to seek an abstract resolution. Mathematics is more focused on constructing a comprehensive understanding rather than the intricate details of arithmetic.

The inclination in mathematics is toward discovering general formulations that can address as many scenarios as possible (a form of reductionism that seeks abstraction and maximizes entropy). In my view, this process is characterized by generalization or abstraction.

The Generalization of Calculus: Level 0 – Telescoping Series

Let’s begin with the simplest iteration of this idea, akin to a “fundamental theorem of series,” though it may not be particularly useful. Nevertheless, it provides a foundation for better understanding certain concepts.

Recall the so-called “useless” Telescoping Series:

Notice how the second term cancels with the third, and the fourth with the fifth, and so forth. In the end, only the first and last terms remain:

In essence, the telescoping series represents a discrete summation where only the extreme values are significant, as the intermediate terms cancel each other out.

The Generalization of Calculus: Level 1 – The Original

Let’s shift from discrete increments to working with infinitesimals (as calculus entails). What if we consider a continuous version of the telescoping series? This leads us directly to the “Fundamental Theorem of Calculus” as we understand it:

At first glance, the connection may not be immediately clear, so let’s break it down.

Initially, in the telescoping series, we know that each middle pair will cancel out. But how can we be sure this holds true in the continuous realm?

We must acknowledge a crucial aspect often required in function analysis: not just continuity, but also differentiability.

Let’s delve into the analytical aspect. The key term is differentiability. This condition guarantees that what is essential in a telescoping series also applies here. When we focus on differentiable (or analytical) functions, we aim to emulate the principle of the telescoping series: in a summation, the intermediate values are irrelevant; only the endpoints matter.

Consequently, the functions that share this property—where middle values cancel out—are termed analytic functions, and the entire foundation of calculus revolves around real analysis, which involves working with analytical functions over real number domains.

We could have easily labeled them “telescoping functions” instead.

A differentiable function is essentially defined as one that possesses derivatives of all orders throughout its domain (where it is defined). The existence of a derivative indicates an internal breakdown akin to a telescoping function: with infinitesimal increments that cancel out when summed.

Thus, the integral identity (and the fundamental theorem of calculus) presented is not universally applicable to all functions. We deliberately focus on functions that satisfy this criterion; these are the analytic functions.

The Generalization of Calculus: Level 2 – Complex Numbers

Once we grasp functions defined on the real number domain, the logical next step is to incorporate imaginary numbers, delving into complex analysis.

Here, we continue working with analytical functions (those that mirror the behavior of the telescoping series at an infinitesimal level).

However, you might wonder: why venture into complex analysis? The answer is that real analysis has its limits. It functions well until it encounters singularities, which complex analysis can address.

The fundamental theorem of calculus, as we know it, tends to falter in the presence of singularities. While it performs adequately without singularities, a singularity can be visualized as a singular point within a 2D space on the complex plane. We can navigate around the singularity and still reach our destination:

The advanced form of the fundamental theorem of calculus manifests as:

The Generalization of Calculus: Level 3 – Multidimensional Analysis

Thus far, we have traversed various mathematical domains; however, multidimensional analysis is not typically a primary focus for mathematicians, although it is crucial in physics. A simple Google search yields limited results for multidimensional analysis compared to real and complex analyses.

Formulas in this area have often emerged not from mathematicians but rather from physicists.

Have you heard of conservative fields? These fields are characterized by work done being independent of the path taken, relying solely on the initial and final points. This concept closely aligns with the fundamental theorem of calculus for analytic functions, emphasizing that only the extreme values—the boundaries—matter, not the chosen path. Therefore, if it forms a closed contour, the work done will be zero:

These outcomes resonate with the closed contour integral of an analytic function (a specific instance of Cauchy’s integral formula):

The only scenario where this does not yield zero is when singularities disrupt continuity.

In one-dimensional analysis, we need only integrate across distances. However, with multidimensional analysis, we must also consider rotations. Thus, we investigate divergence (inner product) and curl (outer product), and as we add more dimensions, complexities increase.

These formulas stem from: Transforming Maxwell’s Equations from Integral to Differential Form (bitdrivencircuits.com).

In physics, when studying electromagnetic fields, we encounter Maxwell’s equations. Since these fields are continuous and differentiable (hence analytical), we can apply the generalization of the fundamental theorem of calculus.

This implies an equivalence between integrating over the entire object or merely its extreme values (or boundaries). Observe the second column:

Physicists have uncovered these laws through experimentation and observation, creating models that align with the data. However, knowing that these fields are analytic, we could have employed the generalized fundamental theorem of calculus to derive them. Still, validation through empirical data is necessary, as fields may contain singularities (if the equality is zero or relates to a charge/mass).

Let’s translate those laws into the language of the fundamental theorem of calculus: The integral over a surface corresponds to integrating over its boundary. This principle also applies to the integration of a volume and its surface boundary.

While it may not be immediately evident from the notation in Maxwell’s equations, it can also be interpreted through the Cauchy integral formula, where charges (or total mass) equate to the residue of the corresponding singularity.

The Generalization of Calculus: Level 4+ – Manifolds

An even broader extension exists beyond the aforementioned concepts, namely manifolds. General Stoke’s theorem (the original integral identity) is fundamentally connected to this idea.

We have progressed from Series, through linear analysis (real analysis), then to curved paths and closed contours (complex analysis), and subsequently to three-dimensional perspectives with divergence and curl (exemplified by Maxwell’s equations). What follows extends into dimensions beyond three, encapsulated within the realm of manifolds.

Manifolds must adhere to the same principles as the telescoping series (real and complex analysis); when we differentiate and sum again, the intermediate values must cancel out, leading to the relevance of only the boundaries.

Conclusion

Ultimately, this is the essence of what constitutes the most beautiful integral: it represents the core of calculus—the fundamental theorem of calculus, extended across multiple dimensions.

All of calculus is built upon this telescoping series behavior. Once differentiation is accomplished, and we sum again, the intermediate values will cancel, leaving only the boundary to consider.

Future Related Work

There seems to be an inconsistency within this narrative: the fundamental theorem of calculus asserts that the sum of small, local changes across a body equals the total global change.

This leads to the conclusion that the closed integral over a body should always yield zero, as it effectively begins and ends at the same point. This holds true for analytical functions but fails in the presence of a singularity. In the case of a meromorphic function, the closed integral produces a value known as the residue, which is not zero. As we sum infinitely small changes, introducing a singularity that represents an “infinite” value alters the results.

A more thorough explanation exists (likely more than one), which involves inner and outer products alongside geometric algebra. I am still a novice in this area, but the differential aspect relates to bivectors. A singularity functions as an outer or something akin.

Regardless, the integral discussed must account for cases involving singularities/residues as well; I just cannot articulate it clearly yet.

Indeed, I believe a singularity elevates the degree of the n-form. Initially, the transformation would be: curve (1-form) > two points (0-form) boundary, but with an internal singularity, it increases by one degree and becomes area (2-form) > curve (1-form) contour. Something like that...

I am still in the early stages of studying differential forms. As I learn more, I anticipate writing about it.

Thank you for reading!