Math 324 is the second half of an introductory multivariable calculus course. The first half is Math 126 (excluding the Taylor series material).
We first study Chapter 15, on the integration of functions of two or three variables. (We will only consider real numbers, not complex, so these are real valued functions of real variables.) That is, we will be integrating functions of the form f: Rn -> R. Here range (or codomain) R is the real numbers, and the domain Rn will usually be the plane R2 or space R3 (or a subset of the plane or space). Visualizing objects and their bounding surfaces in three-dimensional space is one of the biggest challenges in this part of the course.
Next we will jump back to §§14.5-6 to discuss the derivatives of functions f: Rn -> Rm. When m = 1, the derivative can be interpreted as a vector. This leads naturally into §16.1, where we develop a more geometric view of both functions and derviatives by using vector fields.
Chapter 16 combines and builds on ideas from all the chapters 12 through 15 to obtain generalizations of the Fundamental Theorem of Calculus: the Fundamental Theorem for Line Integrals and the Theorems of Green, Stokes, and Gauss. We will assume you remember, or can review quickly, the parametrization of lines and curves in chapter 12 and 13, and learn about line integrals (integrals along curves). These ideas and those in Chapter 15 generalize to parametrization of and integration over planes and curved surfaces in space. (We may do some of the parametrization and integration for surfaces work earlier, to reduce the concentration of key ideas at the end of the course.) The course climaxes with the Theorems, which relate the integral over a surface to an integral around the curve bounding the surface, and an integral over a three-dimensional region (for instance, the interior of a sphere), to an integral over the surface bounding the region.
As you can see, this course is even more "end-loaded" than most math courses. The final weeks build on everything done before, and present some of the most important concepts.
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Most recently updated on September 27, 2011