Calculus III at SPU 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; recognize and apply algebraic and geometric properties of vectors in two and three dimensions; compute dot products and cross products and recognize their geometric meaning; visualize and sketch surfaces in three-dimensional space; compute and interpret partial derivatives of functions of several variables; set up and evaluate double and triple integrals using a variety of coordinate systems, including rectangular, polar, cylindrical, and spherical; use Maple effectively to explore and solve calculus problems; analyze and solve complex problems; write short proofs using the ideas and techniques listed above; and 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
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