Let’s start with what hasn’t changed. The vector calculus chapter will remain largely unchanged.
The earlier chapters, however, have undergone significant revisions. Through several iterations, Nick has developed effective ways to incorporate interactive 3D SageMath graphics that add to student (and instructor) understanding. In addition, the original chapters 9–11 have seen some rewriting and rearrangement, including:
- New Chapter 9 focuses on the precalculus of multivariable functions. We introduce polar (not covered in Active Calculus – Single Variable and vanishing in precalculus curricula), cylindrical, and spherical coordinates in this chapter. Activities in the sections between the introduction of these coordinate systems and using them for integration have students practice working in these coordinate systems. When introducing double or triple integrals in these non-rectangular coordinate systems, the only new topic to address is \(dA\) or \(dV\) in that coordinate system. This chapter also contains vector algebra, lines and planes in \(\mathbb{R}^3\), and a new section on common graphs in three dimensions focusing on cylinder and quadric surfaces that provides additional examples beyond \(z\) as a function of \(x\) and \(y\).
- New Chapter 10 is on the calculus of vector-valued functions (of one variable). Whereas the 1st edition emphasized formalism and algebraic approaches, the 2nd edition will bring geometry and intuition more to the forefront. The material in Chapter 12 of the 1st edition on parameterized curves, which was written purely to support line integrals, has been moved here and updated to reflect its new position.
- New Chapter 11 on derivatives of multivariable functions is organized similar to Chapter 10 from the 1st edition. The 3D graphics have been redone and the narrative and activities place more emphasis on geometric themes. We have also introduced tools that will be used for surface and flux integrals in the vector calculus chapter in places where it is natural to do so, allowing students to have more practice with ideas such as the gradient being normal to a level surface of a function of three variables.
- New Chapter 12 on multiple integrals again follows the outline of Chapter 11 from the 1st edition, but with updates to align its content with the three chapters that precede it, including added emphasis on geometric themes. The section on parametric surfaces and surface area has been moved to Chapter 13, where it fits more naturally into the flow of topics.
- New Chapter 13 on vector calculus sees very few changes from the 1st edition. Essentially, the changes are those that are mentioned above in how other chapters have changed to align with a text that covers the three major theorems of vector calculus.
The 2nd edition of ACM has evolved because in reality, “multivariable calculus” can mean one of three different courses: one that follows chapters 9–11 of the 1st edition and omits vector calculus entirely; one that introduces students to line integrals, the Fundamental Theorem of Calculus for Line Integrals, and Green’s Theorem; and one that proceeds through all of that material plus Stokes’s Theorem and Gauss’s Divergence Theorem. As such, the extent to which certain topics in the new chapters 9–12 are necessary varies depending on the course’s final destination.
The links on this page are to the preview version of the 2nd edition of ACM. We will update this page if further changes are implemented before the official release of the 2nd Edition and update the links to point to the official release after it is made.
