History of differential equations

Differential equations history (overview)
A snapshot overview of the historical origin of differential equations, a mathematical tool invented independently by Isaac Newton (1676) and Gottfried Leibniz (1693). [1]
In mathematics, history of differential equations traces the development of "differential equations" from calculus, itself independently invented by English physicist Isaac Newton and German mathematician Gottfried Leibniz.

The history of the subject of differential equations, in concise form, from a synopsis of the recent article “The History of Differential Equations, 1670-1950”, reads: [2]

Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s.”

Differential equations differ from ordinary equations of mathematics in that in addition to variables and constants they also contain derivatives of one or more of the variables involved.

Newton-Leibniz years
The exact chronological origin and history to the subject of differential equations is a bit of a murky subject; for what seems to be a number of reasons: one being secretiveness, two being private publication issues (private works published only decades latter), and three being the nature of the battle of mathematical and scientific discovery, which is a type of intellectual "war" (in the words of English polymath Thomas Young).

In circa 1671, English physicist Isaac Newton wrote his then-unpublished The Method of Fluxions and Infinite Series (published in 1736), in which he classified first order differential equations, known to hims as fluxional equations, into three classes, as follows (using modern notation): [11]

Ordinary differential equations
Partial differential equations
Class 1Class 2Class 3
Newton 1Newton 2Newton 3

The first two classes contain only ordinary derivatives of one or more dependent variables, with respect to a single independent variable, and are known today as "ordinary differential equations"; the third class involves the partial derivatives of one dependent variable and today are called "partial differential equations".

The study of "differential equations", according to British mathematician Edward Ince, is said to have began in 1675, when German mathematician Gottfried Leibniz wrote the following equation (date of introduction of integral sign; see: symbols):

Leibniz integral (1675)

In 1676, Newton solved his first differential equation. That same year, Leibniz introduced the term “differential equations” (aequatio differentialis, Latin) or to denote a relationship between the differentials dx and dy of two variables x and y. [13]

In 1693, Leibniz solved his first differential equation and that same year Newton published the results of previous differential equation solution methods—a year that is said to mark the inception for the differential equations as a distinct field in mathematics.

Bernoulli years
Swiss mathematicians, brothers Jacob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748), in Basel, Switzerland, were among the first interpreters of Leibniz' version of differential calculus. They were both critical of Newton's theories and maintained that Newton’s theory of fluxions was plagiarized from Leibniz' original theories, and went to great lengths, using differential calculus, to disprove Newton’s Principia, on account that the brothers could not accept the theory, which Newton had proven, that the earth and the planets rotate around the sun in elliptical orbits. [3] The first book on the subject of differential equations, supposedly, was Italian mathematician Gabriele Manfredi’s 1707 On the Construction of First-degree Differential Equations, written between 1701 and 1704, published in Latin. [4] The book was largely based or themed on the views of the Leibniz and the Bernoulli brothers. Most of the publications on differential equations and partial differential equations, in the years to follow, in the 18th century, seemed to expand on the version developed by Leibniz, a methodology, employed by those as Leonhard Euler, Daniel Bernoulli, Joseph Lagrange, and Pierre Laplace.

Integrating factor

In 1739, Swiss mathematician Leonhard Euler began using the integrating factor as an aid to derive differential equations that were integrable in finite form. [12]

Thermodynamics | Condition for an exact differential
See main: Condition for an exact differential; Euler reciprocity relation

The circa 1828 work of English physical mathematician George Green seems to have something to do with defining a test for an “integrable” or conservative field of force (or somehow has connection to thermodynamics via William Thomson); such as in terms of the later 1871 restylized “curl” notation (test of integrability) of James Maxwell (or possibly the earlier work of Peter Tait). [8] In circa 1839, Green stated:

“If all the internal forces exerted be multiplied b the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function.”

The strain-energy potential function of Green is said to of the same theme as Willard Gibbs thermodynamics potentials and Hermann Helmholtz free energy. [9] The use of the both terms “exact differential” and “complete differential” were in common use at least as early as 1841. [7]

From 1850 to 1875, German physicist Rudolf Clausius revolutionized physical science (chemistry, physics, and mechanics) when he transformed the failing notion of French chemist Antoine Lavoisier’s "caloric particle model of heat"—in which a single differential unit or quantity of heat was considered to be an small fluid-like particle (smaller in size than an atom) that was indestructible and said to be located in the interstices of bodies (in the space between the atoms) in various amounts, dependent upon the volume of the given body (more in the body for large volumes; less for smaller volumes) according to Boerhaave’s law—into that of a quantity of heat dQ defined as the product of the absolute temperature T of a body and the “exact differential” quantity entropy dS, such that dQ = TdS, and the physical-mathematical function dQ/T is an extensive exact differential quantity state function. This is probably the most complicated mathematical formalisms in all of human knowledge.

Clausius began to introduce some of the mathematical background to this notion of the "exact differential model of heat" in his 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable”, in which he introduced the now-infamous “condition for an exact differential” to justify his claim that 1/T is the integrating factor (T being the integrating denominator) of the inexact differential function dQ, which makes the resulting function dQ/T an exact differential. The various terminological synonyms and closely related terms are tabulated below:

Exact differential
● A differential equation that satisfies the condition for an exact differential.
● Differential functions of this type, the prime examples (according to the standard model) being state functions, such as entropy dS, enthalpy dH, energy dU, etc., are differential functions that are said to be path independent (in the context of a change of state of a body quantified by the cycle integral, symbol ).
Inexact differential

● A differential equation that does not satisfy the condition for an exact differential.
● Differential functions of this type, the two prime examples being (according to the standard model) differential units of heat  \bar{d}Q \,and work \bar{d}W \,, are differential functions that are said to be path dependent (in the context of a change of state of a body quantified by the cycle integral, symbol ).
Complete differential
Used by Clausius (1858); seems to be a synonym of "exact differential".
Full differential
Seems to be a synonym of "exact differential".
Perfect differential
A rarely used synonym (it seems) for exact differential; found in Spanish versions of thermodynamics.
Imperfect differential

A rarely used synonym (it seems) for inexact differential; found in Spanish versions of thermodynamics.
Total differential
Seems to be a term unrelated to notion of the "complete or exact" differential; the term seems to mean simply the sum of the partial differentials of an equation. In thermodynamics, a "total differential" is not to be confused with a complete or exact differential. [6]
Total exact differential
Is considered a neoplasm; a sort of meaningless term. [6]

This is the sticky point that would go on to make thermodynamics such an immensely difficult subject. The history behind the mathematical concept of the "exact differential" (and corresponding "condition for an exact differential") is in great need of explication.

Candidates for the originator of the notion of the "exact differential" (and "condition for an exact differential") need to be tracked down. In his 1858 article, Clausius mentions the notation usage styles of Swiss mathematician Leonhard Euler (1707-1783) and German mathematician Carl Jacobi (1804-1851). In other parts of his The Mechanical Theory of Heat, Clausius also mentions the work of Irish mathematician William Hamilton (1805-1865).

Other possible or potential candidates for the original formulator of the “condition for an exact differential” may include: Johann Pfaff (1765-1825) or possibly Adrien-Marie Legendre (1752-1833).

American physical economics historian Philip Mirowski seems to think that the notion of the perfect (or "exact") differential arose in the work of Italian mathematician Joseph Lagrange (1736-1813). [5]

Into the 1940s, the terms "exact differential" (vs "inexact differential") were in common use by thermodynamicists, such as Joseph Keenan (1941) and Mark Zemansky (1943).

Notation origin
See main: differential symbols (table)
The English letter d, in the form of dx or dx/dy was introduced in 1675 by German mathematician Gottfried Leibniz. The “curly d” symbol ∂, for partial differential equations was first introduced in 1770 by French mathematician Marquis de Condorcet; then adopted in 1786 by French mathematician Adrien-Marie Legendre; then adopted in 1841 by German mathematician Carl Jacobi, at which point it became the standard.

In 1794, French engineer Lazare Carnot (father of Sadi Carnot), an adherent of the mathematical notation Leibniz, along with French mathematician Gaspard Monge, founded the École Polytechnique, which would go on to become the premier science, engineering, mathematics, and technology school of the early 19th century, and was the first school of thermodynamics, and thus set the theme of future science to use the mathematical notation of Leibniz.

The 1850 to 1875 work of Clausius introduced the mathematical concept of the cycle integral, which later came to be represented by the symbol (although Clausius did not specifically use the integral with the circle in it notation; this seems have been a circa 1920s invention, possibly a notation introduced by English chemical thermodynamicist James Partington).

In 1875, German mathematician Carl Neumann introduced the "d hat" notation ( \bar{d} \,) to represent Clausius' version of the inexact differential an , i.e. one that is path dependent, which is the case with differential units of heat  \bar{d}Q \,and work \bar{d}W \,.

1. Korzybski, Alfred. (1994). Science and Sanity: an Introduction to non-Aristotelian Systems and General Semantics (Section: Differential equations, pg. 595-). Institute of General Semantics.
2. Archibald, Thomas, Fraser, Craig, and Grattan-Guinness, Ivor. (2004). “The History of Differential Equations, 1670-1950”, Mathematrisches Forchungsinstitut Oberwolfach Report 51:2729-94.
3. Tibell, Gunnar. (2008). “The Bernoulli Brothers”, Uppsala University.
4. (a) Manfred, Gabrel. (1707). On the Construction of First Degree Differential Equations (De Constructione Aequationum Differentialium Primi Gradus). Italy.
(b) Manfred Gabriel (Italian → English) – Wikipedia.
5. Mirowski, Philip. (1989). More Heat than Light: Economics as Social Physics, Physics as Nature’s Economics (pg. 31). Cambridge University Press.
6. Perrot, Pierre. (1998). A to Z of Thermodynamics (pg. 105). New York: Oxford University Press.
7. Challis, J. (1841). “A New Method of Investigating the Resistance of the Air to an Oscillating Spring”, Philosophical Magazine (pgs. 229-35). XXXII.
8. Mirowski, Philip. (1989). More Heat than Light: Economics as Social Physics, Physics as Nature’s Economics (pg. 33). Cambridge University Press.
9. Katzir, Shaul. (2006). The Beginnings of Piezoelectricity: a Study of Mundane Physics (pg. 174). Springer.
11. (a) Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
(b) Newton, Isaac. (1964-1967). The Mathematical Works. ed. D.T. Whiteside. Johnson Reprint Corp.
(c) Ince, Edward L. (1926). Ordinary Differential Equations (Appendix A: Historical Note on Formal Methods of Integration, pgs. 529-). Dover.
(d) Sasser, John E. (1992). “History of Ordinary Differential Equations: the First Hundred Years”, Proceedings of the Midwest Mathematics History Society.
(e) Edward Lindsay Ince – Wikipedia.
12. Sasser, John E. (1992). “History of Ordinary Differential Equations: the First Hundred Years”, Proceedings of the Midwest Mathematics History Society.
13. Ince, Edward L. (1926). Ordinary Differential Equations (pg. 3). Dover.

Further reading
● Cajori, Florian. (1928). “The Early History of Partial Differential Equations and of Partial Differentiation and Integration” (abs), The American Mathematical Monthly, 35(9):459-.
● Boole, George. (1859). A Treatise on Differential Equations. MacMillan and Co.
● Forsyth, Andrew R. (1906). Theory of Differential Equations: Part I. Exact Equations and Pfaff’s Problems (exact differential, 4+ pgs). Cambridge University Press.

External links
History of the differential – Math.WPI.edu.
Differential equation – Wikipedia.

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