A snapshot overview of the historical origin of differential equations, a mathematical tool invented independently by Isaac Newton (1676) and Gottfried Leibniz (1693). [1] |

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 equationsbegan 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

Ordinary differential equationsPartial differential equationsClass 1 Class 2 Class 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):

In 1676, Newton solved his first differential equation. That same year, Leibniz introduced the term “differential equations” (

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

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

(clean/add)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:

TermDateDescriptionExact 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 entropydS, enthalpydH, energydU, etc., are differential functions that are said to bepath independent(in the context of a change of state of a body quantified by thecycle integral, symbol ∮). Inexact differential● A differential equation that does notsatisfy 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 and work, are differential functions that are said to bepath dependent(in the context of a change of state of a body quantified by thecycle 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 differentialA rarely used synonym (it seems) for inexact differential; found in Spanish versions of thermodynamics. Total differential Seems to be a term unrelatedto 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

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

The English letterSee main: differential symbols (table)

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

In 1875, German mathematician Carl Neumann introduced the "d hat" notation () to represent Clausius' version of the inexact differential an , i.e. one that is

References

1. Korzybski, Alfred. (1994).

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).

(b) Manfred Gabriel (Italian → English) – Wikipedia.

5. Mirowski, Philip. (1989).

6. Perrot, Pierre. (1998).

7. Challis, J. (1841). “A New Method of Investigating the Resistance of the Air to an Oscillating Spring”,

8. Mirowski, Philip. (1989).

9. Katzir, Shaul. (2006).

11. (a) Newton, Isaac. (c.1671).

(b) Newton, Isaac. (1964-1967).

(c) Ince, Edward L. (1926).

(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).

Further reading

● Cajori, Florian. (1928). “The Early History of Partial Differential Equations and of Partial Differentiation and Integration” (abs),

● Boole, George. (1859).

● Forsyth, Andrew R. (1906).

External links

● History of the differential – Math.WPI.edu.

● Differential equation – Wikipedia.