In what context where vector calculus concepts, such as:

  • Circulation
  • Flux
  • Curl
  • Divergence

first developed? I had assumed they were developed first in fluid dynamics, since the flow of water is concrete and important, and from there applied to electromagnetics, which are obviously more abstract. However, I can find no source to support this.

This post suggests they were first developed for quaternions, ie pure math, and only later applied to physics. My question still stands: What were the initial physical applications and how did they spread?

Related but not duplicate: How were vector calculus nabla ∇ identities first derived?

  • 1
    $\begingroup$ According to Wikipedia, the curl was used for optics by James MacCullagh (1839). Divergence, circulation (as in line integral) and the concept of flux seems to be related to Lagrange surface integrals for fluid mechanics, along with his discovery of the divergence theorem (1762). Another possibility for the origin of flux is Fourier's heat transfer theory (1822). $\endgroup$
    – Mauricio
    Commented Aug 23, 2023 at 22:53
  • 1
    $\begingroup$ Related math.stackexchange.com/questions/139637/… $\endgroup$
    – Mauricio
    Commented Aug 23, 2023 at 22:53
  • $\begingroup$ See How were vector quantities developed? as well as Electromagnetism and vector calculus. $\endgroup$ Commented Aug 24, 2023 at 7:42
  • $\begingroup$ I suggest making a separate post for each of these quantities or making a more general statement like "was vector calculus invented for EM or fluids mechanics?" $\endgroup$
    – Mauricio
    Commented Aug 24, 2023 at 9:17


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