I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?
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2$\begingroup$ Why does it have to be either/or? There were both types of studies, see e.g. Analysis and geometry in the development of the theory of partial differential equations or The History of Differential Equations,1670–1950. $\endgroup$– ConifoldNov 7, 2017 at 21:07
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$\begingroup$ Well it was actually to what extent did it belong to applied mathematics $\endgroup$– Bradley HillNov 8, 2017 at 20:54
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$\begingroup$ @Bradley Hill: 67.5% applied, 32.5% pure. Are you satisfied with such answer? $\endgroup$– Alexandre EremenkoDec 12, 2017 at 1:05
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$\begingroup$ Haha, thank you but I was thinking more 67.55% applied 32.45% pure $\endgroup$– Bradley HillDec 12, 2017 at 10:45
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2 Answers
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It belonged to both. For example, the most important work on PDE in 19th century was arguably Fourier's Analytic theory of heat. You don't have to read the book, to conclude that this was applied mathematics, just from the title. Other important work on PDE, comes from pure mathematics (differential geometry for example), or the work of Liouville on the Liouville's equation.
An interesting example is Cauchy-Riemann equations which come from Complex Analysis (pure mathematics) but closely related to fluid dynamics (applied mathematics).
Laplace's equation appeared first in the work on Saturn rings (applied mathematics) but later was used everywhere and its theory (the theory of harmonic functions) was pure mathematics.
In general, the distinction between pure and applied mathematics was never sharp and clear, especially in 19th century. So the question you are assigned has actually little sense.
answered Nov 7, 2017 at 22:20
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$\begingroup$ I should say it was actually to what extent did it belong to applied mathematics, thank you for your help $\endgroup$ Nov 8, 2017 at 20:56
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I suggest to start with Morris Kline's "Mathematical Thought from Ancient to Modern Times".
A valuable source is also Felix Klein's "Development of Mathematics in the 19th Century".
answered Nov 14, 2017 at 10:25