Wikipedia cites John Anderson’s A History of Aerodynamics and says that velocity potential was introduced by Lagrange in 1788. However, I could trace it at least to Euler 1752, where he published his famous fluid equations. Is it Euler who introduced velocity potential or somebody even before him? It is also possible that Wikipedia is correct and I am misunderstanding something.
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It was Euler c. 1740s, Principia motus fluidorum was read to the Academy in 1752, but published even later. Daniel Bernoulli, who is sometimes also credited, did not introduce not only the velocity potential, but even the "Bernoulli equation" named after him. Lagrange derived it in 1788 from Euler's equations of motion for incompressible fluids, see Takaty, A History and Philosophy of Fluid Mechanics, pp.73-80. Another useful reference is Calero, The genesis of fluid mechanics, 1640-1780, p.38.
However, Bernoulli and Clairaut did give Euler the key ideas in 1743. The notion of total differential appears in Clairaut's Theorie de la Figure de la Terre: Tirée des principes de l'hydrostatique (1743) that Euler was very familiar with, and Bernoulli wrote to him about the elastic potential in a letter the same year. From The Oxford Handbook of the History of Physics, p.366:
"The potential as stored vis viva in an elastic system was first mentioned by Daniel Bernoulli in a letter to Euler of 20 October 1743: ‘For a naturally elastic band, I express the potential live force of the curved band as $\int ds/r^2$ [...] Since no one has perfected the isoperimetric method as much as you, you will easily solve this problem of making $\int ds/r^2$ a minimum.’ Euler proceeded immediately to develop this suggestion in his ‘On the elastic curve’ (1744)... A few years later Euler introduced the potential in fluid dynamics; believing (erroneously) that every stationary flux of an incompressible homogeneous fluid is irrotational, he applied the first elements of differential forms and arrived at ‘Laplace’s equation’: $$ \frac{\partial^2S}{\partial x^2}+\frac{\partial^2S}{\partial y^2}+\frac{\partial^2S}{\partial z^2}=0, $$ where $S$ is the velocity potential (1761a). Moreover, following d’Alembert’s lead in the use of complex numbers in two-dimensional fluid dynamics, Euler (1757c) also introduced what we would now call a complex potential."
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$\begingroup$ Is the complex potential a combination of velocity potential and stream function? Darrigol and Frisch in “From Newton’s mechanics to Euler’s equations” says that the stream function was introduced by D’Alambert in 1761. $\endgroup$– timurCommented Dec 30, 2021 at 18:50
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1$\begingroup$ @timur Stream function and velocity potential give the solenoidal and the irrotational parts of the field, respectively. In 2D this allows to combine them into a holomorphic function, the complex velocity potential. $\endgroup$– ConifoldCommented Dec 31, 2021 at 7:16
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$\begingroup$ I looked at Euler 1757c and having trouble finding where the complex potential appears. I suspect it is an old vs modern notational issue. I will make a new question. $\endgroup$– timurCommented Dec 31, 2021 at 17:08