MATH2730 – Multivariable Calculus (Calculus III)

Text: Calculus (2nd edition), by W. Briggs, L. Cochran, and B. Gillett

Course Delivery: This course is delivered in standard face-to-face lectures with “pencil and paper” homework assignments. However, instructors may need to create a MyMathLab course shell since many students have purchased only the electronic version of the textbook for 1710 (Calculus I). Incidentally, the online textbook has many interactive figures some of which the instructor might wish to demonstrate in the classroom (most notably in chapter 14 on integration in several variables).

Sections which are particularly crucial for later sections and later courses have been underlined.

CH / TITLE/SECTIONS / CONTENT
12 / Vectors and Vector-Valued Functions
12.1 Vectors in the Plane
12.2 Vectors in Three Dimensions
12.3 Dot Products
12.4 Cross Products
12.5 Lines and Curves in Space
12.7 Motion in Space
12.8 Length of Curves
12.9 Curvature and Normal Vectors / This chapter covers the essentials of linear algebra and vector valued functions of one variable which will be needed. Care should be taken to explain the geometric content of both the “dot” and “cross” products (as well as emphasizing that the “dot” is a scalar valued product, while the “cross” is a vector valued one; students often are confused about this point).
Special attention should be paid to the physical interpretations of vector functions. Also, students find curvature and normal vector calculations tedious and difficult at times. Instructors should demonstrate how to do these efficiently.
13 / Functions of Several Variables
13.1 Planes and Surfaces
13.2 Graphs and Level Curves
13.3 Limits and Continuity
13.4 Partial Derivatives
13.5 The Chain Rule
13.6 Directional Derivatives & Gradients
13.7 Tangent Planes and Linear Approximation
13.8 Maximum/Minimum Problems
13.9 Lagrange Multipliers / This chapter and the next are the “core” of the course. This chapter covers differential calculus in several variables, and is analogous to chapters 3 and 4 in 1710. Instructors should take care to explain how the concepts in this chapter are ultimately “the same” as their single variable counterparts, even though the computations are more elaborate.
It should be made clear that the various chain rules in section 13.5 are all manifestations of the same idea. In fact if many of the students are taking (or have taken) 2700 one may wish to at least mention how these can be consolidated using the matrix product.
Students tend to find the method of Lagrange multipliers difficult so plenty of examples should be done here.
14 / Multiple Integrals
14.1 Double Integrals over Rectangles
14.2 Double Integrals over General Regions
14.3 Double integrals in Polar Coordinates
14.4 Triple Integrals
14.5 Triple Integrals in Cylindrical and Spherical Coordinates
14.7 Change of Variable in Multiple Integrals (if time permits) / This chapter covers the evaluation of definite integrals specifically in dimensions 2 and 3. It should be pointed out that the difficulty here shifts from actually computing the integrals (as in 1720) to correctly setting them up (getting the limits of integration correct). Students have difficulty visualizing the regions of integration and consequently typically have problems writing down the limits of integration.
Students should be focused on competence in the “standard” systems (rectangular, cylindrical and spherical). However, if enough time permits 14.7 should be covered (it is more important though to move on to line integrals than spend time here).
15 / Vector Calculus
15.1 Vector Fields
15.2 Line Integrals
15.3 Conservative Vector Fields
15.4 Green’s Theorem / The course should end with at least stating Green’s Theorem and an attempt at imparting its importance even if the section cannot be covered thoroughly (the course is on a pretty tight schedule already). 15.1 can be covered quickly (the concept is not terribly difficult for students). Sections 15.2 and 15.3 need to be covered thoroughly. Students can be overwhelmed by the various (equivalent) formulations of line integrals.