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Über dieses Buch

A half-century ago, advanced calculus was a well-de?ned subject at the core of the undergraduate mathematics curriulum. The classic texts of Taylor [19], Buck [1], Widder [21], and Kaplan [9], for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more signi?cant—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 c- culus on n-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 advancedcalculus course did. Multivariable calculusnaturallysplits intothreeparts:(1)severalfunctionsofonevariable,(2)one function of several variables, and (3) several functions of several variables. The ?rst 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. To put it another way: Green’s theorem ?ts comfortably; Stokes’ and Gauss’ do not.



Chapter 1. Starting Points

Our goal in this book is to understand and work with integrals of functions of several variables. As we show, the integrals we already know from the introductory calculus courses give us a basis for the understanding we need. The key idea for our future work is change of variables. In this chapter, we review how we use a change of variables to compute many one-variable integrals as well as path integrals and certain double integrals that can be evaluated by making a change from Cartesian to polar coordinates.

James J. Callahan

Chapter 2. Geometry of Linear Maps

The geometric meaning of a linear function

$$x \mapsto y = mx$$

is simple and clear: it maps


to itself, multiplying lengths by the factor m. As we show, linear maps


also have their multiplication factors of various sorts, for any


> 1. In later chapters, these factors play a role in transforming the differentials in multiple integrals that is exactly like the role played by the multiplier φ'(


) in the transformation


= φ'(




in single-variable integrals.With this in mind, we take up the geometry of linear maps in the simplest case of two variables.

James J. Callahan

Chapter 3. Approximations

Approximations are at the heart of calculus. In Chapter 1 we saw that the transformation of differentials dx = φ'(s)ds can be traced back to the linear approximation Δx ≈ φ'(s)Δs (the microscope equation), and that the factor φ'(s) represented a local lengthmultiplier.We also suggested there that the transformation

dxdy = rdrdθ

of differentialsfrom Cartesian to polar coordinates has the same explanation: the polar coordinate change map has a linearapproximation (a twovariable “microscope” equation) and the factor


is the local area multiplier for that map. In this chapter we construct a variety of useful approximations to nonlinear functions of one or more variables. However, we save for the following chapter a discussion of the most important approximation, the


of a map.

James J. Callahan

Chapter 4. The Derivative

The derivative of a map is the linear term in its Taylor approximation; it is a map itself. Because linear approximations are simpler than those of higher order, and because linear maps are easier to visualize than nonlinear ones, the derivative is an especially important part of the study of maps. It gives us valuable local information. We study the derivative in this chapter, beginning with the familiar connection to tangents.

James J. Callahan

Chapter 5. Inverses

Inverses help us solve equations: if 5 =


3, then

$$x = \sqrt[3]{5}$$

. Equations also imply relations between their variables. For example, if


2 +


2 -1 = 0, then we can “solve for


” to get either

$$y = + \sqrt {1 - x^2 }$$


$$y = - \sqrt {1 - x^2 }$$

. We soon learn that a formula for an inverse or for an implicitly defined function is seldom available. Usually, the most we can expect to know is that such a function exists. As we show, even this apparently limited knowledge can simplify and clarify our view of a problem, the same way that changing coordinates can simplify an integration. In this chapter, we look only briefly at explicit formulas. We give the bulk of our attention to the way inverses give us a powerful tool for understanding maps, and to the conditions that guarantee their existence. The next chapter does the same for implicitly defined functions.

James J. Callahan

Chapter 6. Implicit Functions

Given a relation between two variables expressed by an equation of the form f (x,y) = k, we often want to “solve for y.” That is, for each given x in some interval, we expect to find one and only one value y = φ(x) that satisfies the relation. The function j is thus implicit in the relation; geometrically, the locus of the equation f (x,y) =k is a curve in the (x,y)-plane that serves as the graph of the function y = φ(x). The problem of implicit functions—and the aim of this chapter—is to determine the function φ from the relation f, or at least to determine that φ existswhen its exact form cannot be found. There are analogues of this problem in all dimensions;that is, x and y can be vectors, and the relation f (x,y) = k can expand intoa set of equations. However, we begin our analysis with a single equation, becausethe various impediments to finding the implicit function already occur there.

James J. Callahan

Chapter 7. Critical Points

At a regular point, the linear terms of a function determine its local behavior,and there is a local coordinate change that transforms the function into one of the new coordinates. At a critical point, the linear terms vanish, but there is still an analogous result for the quadratic terms, called

Morse’s lemma

.However, the quadratic terms may not determine the local behavior, but when they do (the critical point is then said to be


),Morse’s lemma provides a local coordinate change that transforms the function into a sum of positive and negative squares of the new coordinates. In this chapter we analyze Morse’s lemma and use it to characterize critical points.

James J. Callahan

Chapter 8. Double Integrals

Double integrals arise in a variety of scientific contexts, essentially as a way to calculate the product of quantities that vary. They are introduced in the first multivariable calculus course, together with the iterated (repeated) integrals that are often used to evaluate them. This chapter concentrates on definitions and properties,and begins with a problem in gravitational attraction that leads to double integrals. It then introduces a precise notion of area called

Jordan content

, and uses that to define the integral. The next chapter concentrates on evaluation, using iterated integrals,curvilinear coordinates, and the change of variables formula.

James J. Callahan

Chapter 9. Evaluating Double Integrals

Although the definition of the integral reflects its origins in scientific problems, its evaluation relies on a considerable range of mathematical concepts and tools. Most fundamental is the change of variables formula; the single-variable version (“


-substitution”) is perhaps the core technique of integration in the introductory calculus course. By contrast, the method of iterated integrals has no singlevariable analogue; it evaluates a double integral by “partial integration” of one variable at a time. This chapter connects double and iterated integrals, establishes the change of variables formula, and discusses Green’s theorem as a tool for evaluating double integrals and as a reason for orienting them.

James J. Callahan

Chapter 10. Surface Integrals

We turn now to integrals over curved surfaces in space. They are analogous,in several ways, to integrals over curved paths. Both arise in scientific problems as ways to express the product of quantities that vary. The first surface integral we consider measures


, the amount of fluid flowing through a surface. The integrand of a surface integral, like a path integral, can be either a scalar or a vector function: flux is the integral of a vector function, whereas area—another surface integral—is the integral of a scalar. Also, orientation matters, at least when the integrand is a vector function.

James J. Callahan

Chapter 11. Stokes’ Theorem

Stokes’ theorem equates the integral of one expression over a surface to the integral of a related expression over the curve that bounds the surface. A similar result, called Gauss’s theorem, or the divergence theorem, equates the integral of a function over a 3-dimensional region to the integral of a related expression over the surface that bounds the region. The similarities are not accidental. Using the language of differential forms, we show these two theorems are instances (along with Green’s theorem and the fundamental theorem of calculus) of a single theorem that connects one integral over a domain to a related one over its boundary. To explore the connections, we combine the “modern” approach, using differential forms to clarify statements and proofs, with the “classical” appoach, using vector fields to understand the individual theorems in the physical terms in which they arose.

James J. Callahan


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