mathematics

James Eells

James Eells is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand James Eells rather than just read about it. In short: James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis. Biography Eells was born on 25 October 1926, in Cleveland, Ohio.

Key takeaways

James Eells belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect James Eells to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of James Eells from memory before moving on to harder problems.

Reference excerpt

James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis.

Biography Eells was born on 25 October 1926, in Cleveland, Ohio. Eells studied mathematics at Bowdoin College in Maine and earned his undergraduate degree in 1947. After graduation he spent one year teaching mathematics at Robert College in Istanbul and starting in 1948 was for two years an instructor at Amherst College in Amherst, Massachusetts. Next he undertook graduate study at Harvard University, where in 1954 he received his Ph.D. under Hassler Whitney with thesis Geometric Aspects of Integration Theory. In the academic year 1955–1956 he was at the Institute for Advanced Study (and subsequently in 1962–1963, 1972–1973, 1977, and 1982). He taught at Columbia University for several years. In 1964 he became a full professor at Cornell University. In 1963 and in 1966–1967 he was at the University of Cambridge, and after a visit to the new mathematics department developed by Erik Christopher Zeeman at the University of Warwick Eells became a professor of mathematical analysis there in 1969. Eells organized many of the University of Warwick Symposia in mathematics. In 1986 he became the first director of the mathematics section of the Abdus Salam International Centre for Theoretical Physics in Trieste; for six years he served as director in addition to his appointment at the University of Warwick. In 1992 he retired and lived in Cambridge. Eells did research on global analysis, especially, harmonic maps on Riemannian manifolds, which are important in the theory of minimal surfaces and theoretical physics. His doctoral students included John C. Wood. In 1970 he was an invited speaker at the International Mathematical Congress in Nice (On Fredholm manifolds with K. D. Elworthy). He was co-editor of the collected works of Hassler Whitney. Eells's doctoral students include Luc Lemaire, Peter Štefan (1941–1978), Giorgio Valli (1960–1999) and Pierre de la Harpe. Eells was married since 1950 and had a son and three daughters.

Publications Eells, James (1955). "Geometric aspects of currents and distributions". Proceedings of the National Academy of Sciences of the United States of America. 41 (7): 493–496. Bibcode:1955PNAS...41..493E. doi:10.1073/pnas.41.7.493. PMC 528123. PMID 16589704. Eells, James (1959). "On submanifolds of certain function spaces". Proceedings of the National Academy of Sciences of the United States of America. 45 (10): 1520–1522. Bibcode:1959PNAS...45.1520E. doi:10.1073/pnas.45.10.1520. PMC 222748. PMID 16590536. Earle, Clifford J.; Eells, James (1967). "The diffeomorphism group of a compact Riemann surface". Bulletin of the American Mathematical Society. 73 (4): 557–559. doi:10.1090/S0002-9904-1967-11746-4. MR 0212840. with J. H. Sampson: Eells, James; Sampson, J. H. (1964). "Harmonic Mappings of Riemannian Manifolds". American Journal of Mathematics. 86 (1): 109–160. doi:10.2307/2373037. JSTOR 2373037. Eells Jr, James (1966). "A setting for global analysis". Bulletin of the American Mathematical Society. 72 (5): 751–807. doi:10.1090/S0002-9904-1966-11558-6. MR 0203742. Singularities of smooth maps, London, Nelson 1967 with Luc Lemaire: Eells, J.; Lemaire, L. (1978). "A report on harmonic maps". Bulletin of the London Mathematical Society. 10: 1–68. doi:10.1112/blms/10.1.1.; re-published with a follow-up report in the books Harmonic Maps, 1992, and Two Reports on Harmonic Maps, 1994, by publisher World Scientific with Luc Lemaire: Selected topics in harmonic maps, AMS 1983 with Andrea Ratto: Harmonic maps and minimal immersions with symmetries – methods of ordinary differential equations applied to elliptic variational problems, Princeton University Press 1993 with B. Fuglede: Harmonic maps between Riemannian polyhedra, Cambridge University Press 2001

Illustrations

Worked examples

Example 1 — a first encounter with James Eells

Start with the simplest possible case. Write down what James Eells claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to James Eells before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about James Eells ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of James Eells

In research: James Eells appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses James Eells in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: James Eells is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1926 births, 2007 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for James Eells outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study James Eells in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what James Eells means in your own words.
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Work through the three examples above with pen and paper.
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Frequently asked questions

What is James Eells in simple terms?

James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis. Biography Eells was born on 25 October 1926, in Cleveland, Ohio.

Why does James Eells matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study James Eells?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on James Eells.