Newton's Calculus Notes

The calculus was independently discovered by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, marking a pivotal moment in the history of mathematics.

This discovery was not merely about finding effective methods for solving tangent and quadrature problems; such problems had been explored before by mathematicians like Fermat and others.

Earlier solutions to these problems were often ad hoc, involving special methods tailored to particular cases, and lacked the generalized algorithmic procedures that calculus would provide.

The true significance of Newton and Leibniz's work lies in their recognition of a fundamental relationship and their ability to use it as a foundation for further progress. This often involved introducing new terminology, notation, and frameworks.

Pierre de Fermat, for instance, constructed the expression f(A+E) - f(A), observed E as a factor, divided by E, and canceled terms containing E. However, he did not formalize this into a general method or introduce specific notation.

Had Fermat formalized his methods and introduced a consistent notation, he might be considered a co-discoverer of differential calculus, highlighting the importance of formalization in mathematical discovery.

The "fundamental theorem of calculus" establishes the inverse relationship between finding tangents to curves (differentiation) and finding areas under curves (integration). This theorem is the cornerstone of calculus.

This inverse relationship was implicit in earlier area computations. For example, the area under the curve y = x^n from [0, x] is \frac{x^{n+1}}{n+1}, and the tangent slope of the curve y = \frac{x^{n+1}}{n+1} is x^n. This exemplifies the inverse relationship.

Isaac Barrow, Newton's mentor, stated and proved a geometric theorem describing this inverse relationship but did not recognize its potential as the basis for a new, systematic method of calculation.

Newton and Leibniz not only recognized the fundamental theorem but also harnessed its power to create a potent algorithmic instrument for systematic calculation, thus revolutionizing mathematics and discovering calculus.

Isaac Newton (1642-1727) was born on Christmas Day in 1642 and made profound contributions to mathematics, physics, and astronomy.

He entered Cambridge University in 1661 and, in 1669, succeeded Barrow as the Lucasian Professor of Mathematics, a position he held until 1696. He then left for London to become Warden of the Mint.

Newton was buried in Westminster Abbey upon his death in 1727, a testament to his esteemed status.

Newton began his serious study of mathematics in the summer of 1664, starting with Euclid's Elements and Descartes' Geometrie, which laid the groundwork for his later achievements.

During the closure of Cambridge due to the plague in 1665-1666, Newton, in what he termed his biennium mirabilissimum (year of wonders), laid the foundations for calculus, explored the nature of light, and formulated his theory of gravitation. These years were remarkably productive.

Newton's seminal works, Principia Mathematica (1687) and Opticks (1704), covered mechanics and optics, respectively. However, many of his pure mathematics contributions remained largely unpublished during his lifetime.

Newton often communicated his mathematical discoveries through letters and manuscripts rather than formal publications in journals, a common practice at the time.

Newton left behind approximately 5000 sheets of unpublished mathematical manuscripts, which have since been studied and compiled.

D.T. Whiteside undertook the monumental task of editing The Mathematical Papers of Isaac Newton, resulting in an eight-volume collection published from 1967.

"The October 1666 Tract on Fluxions" represents Newton's first formal paper on calculus ([NP I], pp. 400–448), providing valuable insights into his early work.

Newton approached the tangent problem by considering velocity components of a moving point, an approach reminiscent of Roberval's earlier investigations.

He conceptualized a curve f(x, y) = 0 as the locus of the intersection of vertical and horizontal moving lines, providing a dynamic way to understand curves.

In Newton's framework, coordinates x and y are functions of time t, and motion along the curve is a composition of horizontal and vertical motions.

The tangent velocity vector is derived from the parallelogram sum of horizontal (\dot{x}) and vertical (\dot{y}) vectors, and the slope of the tangent line is given by the ratio \frac{\dot{y}}{\dot{x}}.

Newton envisioned points A and B moving distances x and y along straight lines in equal time, subject to the condition that f(x, y) = 0 always holds.

Although Newton did not explicitly define "fluxional speeds," he treated the speed of a point moving along a straight line as an intuitively understood concept.

In modern terminology, fluxions \dot{x} and \dot{y} are equivalent to derivatives with respect to time t, such that \dot{x} = \frac{dx}{dt} and \dot{y} = \frac{dy}{dt} . Their ratio gives the derivative of y with respect to x, expressed as \frac{\dot{y}}{\dot{x}} = \frac{dy}{dx}.

Initially, Newton employed letters like p and q to represent fluxions, but he later adopted the dot notation (\dot{x} and \dot{y}) consistently in the early 1690s.

Newton's initial problem was to determine the relationship between fluxions \dot{x} and \dot{y}, given the equation f(x, y) = 0. For a polynomial f(x, y) = \Sigma a_{ij}x^iy^j, the solution involves multiplying each term by \frac{\dot{x}}{x} according to the x dimensions and by \frac{\dot{y}}{y} according to the y dimensions ([NP I], p. 402).

Thus, if f(x, y) = \Sigma a{ij}x^iy^j = 0, then \Sigma (i\frac{\dot{x}}{x} + j\frac{\dot{y}}{y})a{ij}x^iy^j = 0 (1).

Exercise 1 demonstrates that equation (1) is equivalent to \dot{x}\frac{\partial f}{\partial x} + \dot{y}\frac{\partial f}{\partial y} = 0, and consequently, \frac{\dot{y}}{\dot{x}} = - \frac{\partial f / \partial x}{\partial f / \partial y}.

Newton observed that distances traversed are proportional to velocities, even when velocities are not uniform during an infinitesimally short time interval $$\omicron