I noticed there are two varieties of Maxwell's equations and although I am not sure which form Maxwell originally worked with and perhaps the differential form came much latter, I was wondering if he made use of Green's Theorem. I ask because after reading about Green , I got the impression he was motivated to develop his theorem for trying to understand electrical behavior. I looked at Wiki and James Clerk Maxwell lived from 1831 to 1879 and George Green from 1793 to 1841. Certainly Maxwell was old enough to have read about Green's work before Green passed away.
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$\begingroup$ I don't understand the last sentence. Are you saying that Maxwell was reading research-level physics while still in primary school? $\endgroup$– Peter TaylorCommented Oct 10, 2017 at 16:19
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$\begingroup$ Well he was 23 when he took the exam sited below. $\endgroup$– SedumjoyCommented Oct 11, 2017 at 19:47
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Of course Maxwell knew Green's theorem, by the time he was writing this was the common knowledge. Maxwell's book has a mathematical preliminary chapter (chapter 2) where he explains mathematical tools he uses, and this contains Gauss, Green, Stokes theorems and much more. (In fact he anticipates what was later called Hodge theory). In the chapter where Maxwell equations are derived, (Chap 9, vol. 2) he freely passes from differential to integral formulations and back.
Green's book (1828) remained in obscurity for some time, until William Thomson (future Lord Kelvin) found it in 1845 and made his results widely known. And yes, Green's book was called "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism", so it is about electricity and magnetism.
answered Oct 8, 2017 at 13:12
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$\begingroup$ So they each had a piece what Maxwell needed and Maxwell was genius enough to put them all together in a way to explain the physical observations. $\endgroup$– SedumjoyCommented Oct 8, 2017 at 22:35
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1$\begingroup$ @Sedumjoy: Every scientist uses the tools developed by his predecessors. Maxwell was never credited with these pure mathematical formulas. His great discoveries in electromagnetism certainly cannot be described as "putting together" what was known before. I advise you to read what Maxwell really did. $\endgroup$ Commented Oct 9, 2017 at 13:04
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2$\begingroup$ @Sedumjoy: His main discoveries concerning electromagnetism belong to physics rather than mathematics: they are the displacement current (nobody even conjectured that it exists), and as a result, Maxwell equations in their present form, and identification of light with electromagnetic waves. This was probably the greatest discovery in physics since Newton. But the necessary mathematical tools needed for this were already developed by that time. $\endgroup$ Commented Oct 9, 2017 at 13:12
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2$\begingroup$ In a footnote on p. 27 of his book on Electricity and Magnetism, Maxwell states that Stokes posed what is now known as Stokes's theorem (surface case) as problem 8 of the 1854 Smith's Prize Examination (which Maxwell took). The full exam can be found at www.clerkmaxwellfoundation.org/SmithsPrizeExam_Stokes.pdf Maxwell had the high score on the exam and obviously must have solved the problem in question. As Alexandre Eremenko states, in that same book Maxwell attributes several results to Green and uses the phrase "Green's Theorem" as if it were already well established terminology. $\endgroup$– Dan FoxCommented Oct 9, 2017 at 18:32
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2$\begingroup$ @Dan Fox: thanks for finding this famous exam! I wonder how many graduate students today can do it:-) $\endgroup$ Commented Oct 9, 2017 at 18:34