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A half-century ago, advanced calculus was a subject at the core of the undergraduate mathematics calculus. The classic texts of Taylor and , for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more those seen as giving a rigorous foundation to calculus and they – came the basis for a new course, an introduction to real analysis, that eventually supplanted advanced calculus in the core. Advanced calculus did not, in the process, become less important, but its role in the curriculum changed. In fact, a bifurcation occurred. In one direction we got calculus on manifolds, a course beyond the practical reach of many undergraduates; in the other, we got calculus in two and three dimensions but still with the theorems of Stokes and Gauss as the goal. The latter course is intended for everyone who has had a year-long introduction to calculus; it often has a name like Calculus III. In my experience, though, it does not manage to accomplish what the old advanced calculus course did. Multivariable calculus naturally splits into three parts:(1)several functions of on available,(2)one function of several variables, and (3) several functions of several variables. The two are well-developed in Calculus III, but the third is really too large and varied to be treated satisfactorily in the time remaining at the end of a semester.
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