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Calculus III

Revised: September 2020

Course Description

Plane curves, polar coordinates, vectors and solid analytic geometry, vector-valued functions, partial differentiation, multiple integrals. Prerequisite: MATH 255. Four semester hours.

Student Learning Objectives

By the end of the course students should be able to:

• Work with and visualize graphs of functions of several variables;
• Geometrically and algebraically describe vectors and vector operations;
• Differentiate and integrate vector valued functions, and using these operations appropriately in applications;
• Differentiate multivariate functions and determine when partial differentiation or ordinary differentiation is needed;
• Use differentiation, directional derivatives, and the gradient in solving applied problems;
• Solve constrained and unconstrained optimization problems with several independent variables;
• Setup and evaluate double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates; and
• Work with parametric curves, equations, and vector fields.

Text

Gilbert Strang, Edwin Herman, et al., Calculus, Volume 3, OpenStax, 2018

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• CHAPTER 1 -Parametric Equations and Polar Coordinates (1 week)
  • Parametric equations, calculus of parametric curves
• CHAPTER 2 -Vectors in Space (2 weeks)
  • Vectors in the Plane, Vectors in 3 Dimensions, Dot Product, Cross Product, Equations of Lines and Planes in Space, Quadric Surfaces

• CHAPTER 3 -Vector-Valued Functions (1.5 weeks)
  • Vector-Valued Functions and Space Curves, Calculus of Vector-Valued Functions, Arc Length and Curvature, Motion in Space

• CHAPTER 4 -Differentiation of Functions of Several Variables (3.5 weeks)
  • Functions of Several Variables, Limits and Continuity, Partial Derivatives, Tangent Planes and Linear Approximations, Chain Rule, Directional Derivatives and Gradient, Maxima/Minima Problems, Lagrange Multipliers    
• CHAPTER 5 -Multiple Integration (2 weeks)
  • Double Integrals over Rectangular Regions and General Regions, Double Integrals in Polar Coordinates, Triple Integrals, Triple Integrals in Cylindrical and Spherical Coordinates, Calculating Centers of Mass and Moments of Inertia (optional), Change of Variables in Multiple Integrals

• CHAPTER 6 -Vector Calculus (as time allows)
  • Vector Fields, Line Integrals, Conservative Vector Fields, Green’s Theorem, Divergence and Curl, Surface Integrals, Stokes’ Theorem