Was it unusual for people in those days to learn Calculus? Could a grad student take a course in differential equations or multi-variable Calculus, or did they have to learn from journals? I am always amazed with famous people like Riemann. I tried to get a grasp of analytic number theory, but it's just too hard.
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3$\begingroup$ As a note: archives of old Tripos examinations might give a clue as to what was expected of students in the olden days... $\endgroup$– J. M. ain't a mathematicianCommented Feb 13, 2012 at 0:26
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7$\begingroup$ I get the impression that today, calculus plays the same role that Greek did in Britain in that era. They wanted to keep the riffraff from being, e.g., military officers, so they required them to know Greek. Today, premeds in the University of California system have to take calculus-based physics. Ca. 1900, doctors in California didn't typically have college degrees at all. Vector calculus wasn't invented until ca. 1850, and the modern notation didn't exist until the 20th century. Einstein's 1905 paper on relativity doesn't use div, grad, or curl, making it horrible for modern readers to read. $\endgroup$– user13618Commented Feb 13, 2012 at 1:09
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It was not unusual for a student at the University of Cambridge to learn calculus.
If you look at a Tripos paper for 1786 (reproduced on page 183 of Mathematical Recreations and Essays by W. W. Rouse Ball, available from Project Gutenberg) there were relatively simple questions on fluxions and fluents.
By the end of the 19th century, Cambridge had gone well beyond this: two of the fourteen (unreformed) 1906 Part I papers can be found at Project Euclid (or this AMS link).
A wider review can be found in Mathematics in Victorian Britain by Raymond Flood, Adrian Rice, Robin Wilson, and extracts can be seen by following this link from Google Books.
The current concept of a graduate student is rather modern. Those seeking an academic career would have taken other roles such as lecturers and tutors while seeking a college or university fellowship. In the early period, most in England would have been ordained Anglican priests (as Charles Dodgson/Lewis Carroll was) and used their spare time for research.
Sources would have included journals, books and correspondence. A researcher had greater personal responsibility for their own efforts than they do now, but since less research by others existed then, it was easier to keep up-to-date.
