Introduction
Few developments in the rich history of mathematics may exult in advancing our comprehension and perception of the world as can the advent of calculus. Calculus germinated from the mathematical loam of mensuration, nourished by the burgeoning concerns of physics. The problems of calculus were tackled as early as Eudoxus of Cnidus and Archimedes, who had developed the method of exhaustion to compute areas and volumes by approximating figures with sequences of inscribed and circumscribed polygons, proving results through double reductio ad absurdum arguments. The notion of infinitesimals although spurned for its apparent lack of rigour, nonetheless proved an essential heuristic as evinced by Archimedes’ Method of Mechanical Theorems who envisioned plane figures as comprised of indivisible slices to obtain an ansatz before applying the method of exhaustion. Limit-esque arguments also appeared in fifteenth-century Kerala, where Madhava of Sangamagrama and his successors, obtained infinite series for the sine, cosine, and arctangent functions by subdividing an arc into increasingly many small segments, summing associated geometric quantities, and passing to the limit with infinitely many segments. The critical breakthrough came as the epiphany that the problems of quadrature and tangents were entwined. James Gregory and Isaac Barrow articulated special cases of this reciprocity geometrically. Soon, Newton and Leibniz developed a unification of the concepts of area under the curve and instantaneous rate of change embodied in Theorem 1.
Theorem 1 (Newton-Leibniz) Let \(f\) be an integrable real-valued function and \(g,h\) be differentiable real-valued functions then: \[ \frac{d}{dx}\int_{h(x)}^{g(x)}{f(t) dt} = f(g(x))g'(x) - f(h(x))h'(x) \]
Remark. Seen differently, taking \(h(x) = a, g(x) = x\) and sending \(f \to f'\) we have \(\frac{d}{dx}\bigl(\int_{a}^{x}{f'(t) dt} - f(x)\bigr) = 0\) which is a differential equation with only constant solutions, and since \(\int_{a}^{a}{H(t)dt}=0\) for any \(H\), we have: \(\int_{a}^{x}{f'(t) dt} = f(x) - f(a)\) and evaluating at \(x = b\) we get: \[ \int_{a}^{b}{f'(x) dx} = \int_{a}^{b}{\frac{df}{dx} dx} = \int_{[a,b]}{df} = f|^{b}_{a} \] Setting aside, tentatively, the “cancellation” of \(dx\) as mere notational convenience, latter equality expresses the quantities under concern in a coordinate-free manner on the object integrated over (the interval). Indeed, we may write that Theorem 1 tells us that integrating \(df\) over an elementary set (union of finitely many intervals) in \(\mathbb{R}\) boils down to an alternating sum of evaluations of \(f\) on the boundary points.
Corollary 1 (Multi-variable variant) Let \(f: \mathcal{D} \to \mathbb{R}\) be a scalar field over \(\mathcal{D} \subseteq \mathbb{R}^n\), and be any curve \(C\) with a differentiable parametrization \(\gamma: [0,1] \to \mathbb{R}^n\) then: \[ \int_{C}{\nabla f \cdot d\mathbf{l}} = \int_{0}^{1}{\nabla f(\gamma(t)) \cdot \mathbf{\gamma}'(t) dt} = f \circ \gamma|^{1}_{0} \]
Remark. Follows from \(\frac{d}{dt}(f \circ \gamma)\) and the “law of total derivative”. Alternatively, by first principles, one can re-write the difference quotient involved in the derivative as the sum of difference quotient involving single-variable moves.
Excluding seminal results, the resemblance between the calculus of Newton and the elementary calculus of today is superficial at best, with the former owing much to an intuitive use of infinitesimals which was evidently inconsistent to such an extent that the immaterialist philosopher George Berkeley likened them to religious doctrine his publication “The Analyst: A Discourse Addressed to an Infidel Mathematician”. Posterity is witness to the salubrity of such works in impelling the “arithmetization” of calculus under Cauchy and Weierstrass, reforming the system under the stalwart concepts of limits, \(\epsilon\delta\)-proofs and real analysis. Despite criticism, the relentless quest of physics to understand the universe had not been delayed for the want of analysis, having already tasted the fruits of this “doctrine” in mechanics, and Newton’s theory of gravitation.
The study of heat conduction, elasticity, gravitation, fluid flow and electromagnetism in the eighteenth and nineteenth centuries required a calculus of vector fields defined over regions of space, and specifically demanded a way to relate quantities inside a region to its behavior on the region’s boundary. Gauss’ laws, which reformulate inverse-square laws for gravitational and electric fields relating flux through a surface directly to an enclosed quantity such as mass or charge, are prime examples. Gauss’ divergence theorem (a weakening of which is Lemma 1), Green’s theorem and Stokes’ theorem grew out of this motivating body of work in a short period of time.
Lemma 1 Let \(R: \mathbb{R}^3 \to \mathbb{R}\) be a differentiable scalar field and \(V = \{(x,y,z) \in \mathbb{R}^3 | (x,y) \in \mathcal{D}, 0 \leq z \leq f(x,y)\}\) be the volume enclosed by the graph of \(f\) over \(\mathcal{D}\), with surface \(\partial V\) then: \[ \int_{V}{\frac{\partial R}{\partial z} dV} = \int_{\partial V}{R \hat{\mathbf{k}} \cdot d\mathbf{S}} \]
Proof. The convention for surface integrals is to chose the outward pointing normal. For the surface defined by \(f\) we get normals: \(\mathbf{n} = -\frac{\partial f}{\partial x}\hat{\mathbf{i}} -\frac{\partial f}{\partial y}\hat{\mathbf{j}} + \hat{\mathbf{k}}\) \[ \iint_{D}{\int_{0}^{f(x,y)}{\frac{\partial R}{\partial z} dz} dy dx} = \iint_{D}{R(x, y, f(x,y)) - R(x,y, 0) dy dx} = \int_{S}{R\hat{\mathbf{k}}\cdot d\mathbf{S}} + \int_{-P}{R\hat{\mathbf{k}}\cdot d\mathbf{S}} \] where \(S\) is the surface defined by \(f\), and \(P\) is the xy-plane restricted to \(\mathcal{D}\) with orientation such that normal is upward. It is clear that vertical faces in \(\partial V\) contribute nothing to the surface integral, so the lemma is established.
Remark. We may conclude that Lemma 1 holds for all unidirectional vector fields since the quantities under consideration are independent of coordinates and we may always choose rectilinear coordinates \((x, y, z)\) such that the field is aligned with \(\hat{\mathbf{k}}\). Additionally, we can extend the result to volumes enclosed by two surfaces (given by \(f_1\), \(f_2\)) glued on their common edge by applying Lemma 1 on \(R, f_1\) and \(-R, -f_2\) and summing: \[ \int_{S_1}{R \hat{\mathbf{k}} \cdot d\mathbf{S}} + \int_{-P}{R \hat{\mathbf{k}} \cdot d\mathbf{S}} + \int_{-S_2}{-R \hat{\mathbf{k}} \cdot d\mathbf{S}} + \int_{-P}{-R \hat{\mathbf{k}} \cdot d\mathbf{S}} =\int_{S_1 + S_2}{R \hat{\mathbf{k}} d\mathbf{S}} \] Indeed, we can take this a step further, noticing that carving out a cuboid in the interior of the volume leaves the flux unchanged, and that the lemma readily applies to cuboids. So, for a closed surface formed by “gluing together” graphs, the theorem applies. A general vector field can be broken down into orthogonal components, and thus Gauss’ theorem for divergence follows by linearity of divergence for such 3D surfaces:
\[ \int_{V}{\nabla \cdot \mathbf{D} dV} = \int_{\partial V}{\mathbf{D} \cdot d\mathbf{S}} \]
Cauchy, three years prior to Green, had begun investigations into Complex Analysis, and discovered the celebrated Cauchy’s Formula for Contour integrals, based on a key result:
Lemma 2 (Cauchy, 1825) Let \(\mathcal{D} \subseteq \mathbb{C}\) be the interior of a contour \(C\), such that \(\forall z \in \mathcal{D}, f \text{ complex-differentiable at } z\) then: \[ \oint_{C}{f(z) dz} = 0 \]
Remark. The superficial similarity between this beautiful result of Cauchy and that of Corollary 1 for a closed loop is tantalizing. This is yet another instance of the behaviour of a function on the interior determining boundary phenomenon.
Gauss’s Disquisitiones generales circa superficies curvas first demonstrated, in 1827, that a surface’s curvature could be characterized entirely by measurements internal to the surface itself, independent of ambient space. Riemann generalized this intrinsic viewpoint to spaces of arbitrary dimension equipped with a metric, supplying the notion of “space” abstracted entirely from any embedding. Hermann Grassmann’s exterior algebra furnished the algebraic apparatus of alternating multilinear forms and wedge products. Élie Cartan synthesized Grassmann’s exterior algebra, Riemannian geometry, and the emerging theory of smooth manifolds, to reformulate the differential geometry of curved spaces in terms of exterior calculus, motivated in no small part by the demands of General Relativity to be rid of the grime of coordinate-based descriptions.
The condensed history of calculus presented omits, by necessity of brevity, doubtless multifarious developments such as alternative energy-based methods of Lagrangian mechanics that increasingly drove the need to apply calculus to “higher-order objects” such as functionals, leading to Calculus of Variations or the formalization of infinitesimals in hyperreals under Non-Standard Analysis. The purpose of such a long-winded presentation is to highlight among these gems of calculus, the recurring theme of being able to move from the boundary to interior as the domain of integration while swapping the integrand to a corresponding term with one more “order of derivative”. These gems of calculus, although related by this common theme do not overtly appear to be connected, yet the notion of behaviour on the boundary being determined by variation within the interior prevails intuitively, instilling a desire for their unification under an elegant framework that formalizes this intuition.
This article is unusual in its design, in that it is devoted exclusively to the presentation, as juxtaposed with the development, of the celebrated crown jewel of this body of work due to Cartan, known today as the Generalized Stokes’ Theorem, with the sole purpose of inspiring an intuitive understanding of this beautiful result, solidifying its connections with familiar facts, and obtaining an operational facility in applying it. The diligent reader is referred to the numerous established texts for the purpose of attaining a pragmatic, flexible and deep working knowledge of this fundamental theorem of calculus.