SPU Mathematics

Course Objectives for MAT 1236: Calculus III

This third course in calculus continues the study of differential and integral calculus begun in MAT 1234 and 1235.  The course includes parametric equations, polar coordinates, vectors, sequences, series, and Taylor expansions. It also introduces multivariable calculus, including partial derivatives, double integrals, and triple integrals.  The primary aims of the course are to help students develop new problem solving and critical reasoning skills and to prepare them for further study in mathematics, the physical sciences, or engineering.  By the end of the course, students should be able to

• sketch and analyze curves given parametrically;

• graph curves in polar coordinates;

• compute areas and arc lengths using rectangular and polar coordinates;

• understand the algebra and geometry of vectors in two and three dimensions;

• compute dot products and cross products and understand their geometric meaning;

• visualize and sketch surfaces in three-dimensional space;

• find partial derivatives of functions of several variables and demonstrate an understanding of their meaning;

• set up and evaluate double and triple integrals using a variety of coordinate systems, including rectangular, polar, cylindrical, and spherical; and

• use Maple to effectively to explore and solve calculus problems.

In addition to the specific skill-oriented objectives above, students should

• have improved skills at problem solving and critical thinking: at dissecting a complex problem, determining steps in its solution, finding the solution, and testing whether it is reasonable;

• be able to write short proofs using the ideas and techniques listed above; and

• be able to provide clear written explanations of the ideas behind key concepts from the course.

Students should also gain an increased appreciation of mathematics as part of the language of science and as a study in itself.

Course Content

The standard material to be covered in Calculus III from the 6th edition of Stewart's Calculus is listed below.  Individual instructors may make minor modifications to this list.

• 11.1 Curves Defined by Parametric Equations

• 11.2 Calculus with Parametric Curves

• 11.3 Polar Coordinates

• 11.4 Areas and Lengths in Polar Coordinates

• 12.1 Sequences

• 12.2 Series.

• 12.3 The Integral Test and Estimates of Sums.

• 12.4 The Comparison Tests.

• 12.5 Alternating Series.

• 12.6 Absolute Convergence and the Ratio and Root Tests.

• 12.7 Strategy for Testing Series.

• 12.8 Power Series.

• 12.9 Representation of Functions as Power Series.

• 12.10 Taylor and Maclaurin Series.

• 12.11 Applications of Taylor Polynomials

• 13.1 Three-dimensional Coordinate Systems

• 13.2 Vectors

• 13.3 The Dot Product

• 13.4 The Cross Product

• 13.5 Equations of Lines and Planes

• 13.6 Quadric Surfaces

• 15.1 Functions of Several Variables

• 15.2 Limits and Continuity

• 15.3 Partial Derivatives

• 15.5 The Chain Rule

• 16.1 Double Integrals over Rectangles

• 16.2 Iterated Integrals

• 16.3 Double Integrals over General Regions

• 16.4 Double Integrals in Polar Coordinates

• 16.5 Applications of Double Integrals

• 16.6 Triple Integrals

• 16.7 Triple Integrals in Cylindrical Coordinates

• 16.8 Triple Integrals in Spherical Coordinates