Putting differentials back into calculus - Department of Mathematics ...

discussing linear approximation, the door would be left open for those who so chooseto use differentials to do calculus, with all of the advantages we have described.Science is about physical quantities, not functions. Mathematicians justifiably pridethemselves on the development of a rigorous calculus of functions, but it is rather thecalculus of physical quantities which is relevant to scientists. Whether one regardsdifferentials as the conceptual core of calculus, as we do, or merely a heuristic tool,they reflect the way many of our professional colleagues think about calculus. Thisinvaluable tool should be emphasized, not relegated to a recently-invented role in linearapproximation, for which it is neither suited nor needed.Surely we should encourage the teaching of a concept which captures the intuitiveessence of calculus, which is easy for students to master, and which always producescorrect answers.Acknowledgment. This work was supported in part by NSF grant DUE–0618877.Summary. We argue that the use of differentials in introductory calculus courses providesa unifying theme which leads to a coherent view of calculus. Along the way, we will meetseveral interpretations of differentials, some better than others.References1. C. B. Allendoerfer, Differentials (editorial), Amer. Math. Monthly 59 (1952) 403–406; reprinted in [2].2. T. M. Apostol, et al., eds., Selected Papers on Calculus, Mathematical Association of America, WashingtonDC, 1969.3. J. Bair and V. Henry, From mixed angles to infinitesimals, College Math. J. 39 (2008) 230–233.4. G. Berkeley, The analyst, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics,Vol.1,William Ewald, ed., Oxford University Press, Oxford, 1996; reprint of pamphlet distributed in 1734; availableat http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf.5. A. Church, Differentials, Amer. Math. Monthly 49 (1942) 389–392; reprinted in [2]. doi:10.2307/23031366. R. Dawson, Differentiate early, differentiate often!, College Math. J. 36 (2005) 404–407.7. T. Dray and C. A. Manogue, The vector calculus gap, PRIMUS 9 (1999) 21–28; available at http://www.math.oregonstate.edu/bridge/papers/calculus.pdf. doi:10.1080/105119799089659138. , Conventions for spherical coordinates, College Math. J. 34 (2003) 168–169; available at http://www.math.oregonstate.edu/bridge/papers/spherical.pdf.9. , Using differentials to bridge the vector calculus gap, College Math. J. 34 (2003) 283–290; availableat http://www.math.oregonstate.edu/bridge/papers/use.pdf. doi:10.2307/359576510. H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, 1963;reprinted by Dover, Mineola NY, 1989.11. M. K. Fort, Jr., Differentials, Amer. Math. Monthly 59 (1952) 392–395; reprinted in [2]. doi:10.2307/230681412. D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Prentice-Hall, New York, 1999.13. M. Kac and J. F. Randolph, Differentials, Amer. Math. Monthly 49 (1942) 110–112; reprinted in [2]. doi:10.2307/230266614. H. J. Keisler, Elementary Calculus: An Infinitesimal Approach, 2nd ed., PWS, Boston, 1982; available athttp://www.math.wisc.edu/~keisler/calc.html.15. G. Leibniz, New method for maximums and minimums, in A Source Book in Mathematics, 1200–1800, DirkJan Struik, ed., Harvard University Press, Cambridge MA, 1969; translation of Nova methodus pro maximiset minimis, pamphlet distributed in 1684.16. W. G. McCallum, review of Calculus Mysteries and Thrillers by R. G. Woods and How To Ace Calculus: TheStreetwise Guide by C. Adams, J. Haas, and A. Thompson, Amer. Math. Monthly 108 (2001) 90–93. doi:10.2307/269570017. W. McCallum, D. Hughes-Hallett, A. Gleason, et al., Calculus, 4th ed., Wiley, Hoboken NJ, 2005.18. C. G. Phipps, The relation of differential and delta increments, Amer. Math. Monthly 59 (1952) 395–398;reprinted in [2]. doi:10.2307/230681519. W. R. Ransom, Bringing in differentials earlier, Amer. Math. Monthly 58 (1951) 336–337; reprinted in [2].doi:10.2307/2307725VOL. 41, NO. 2, MARCH 2010 THE COLLEGE MATHEMATICS JOURNAL 99