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Calculus III
Calculus III covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integrations, and second-order differential equations.
Each module begins with a prerequisite material review section, in which critical concepts from Precalculus, College Algebra, Calculus I, and Calculus II are revisited. Additionally, just-in-time reviews of essential math concepts appear throughout the text to help those students who need further learning support. The course includes embedded, algorithmically generated practice questions, worked-example videos, and a complete set of outcome-aligned online assessments in OHM.
For even more practice opportunities, there are problem sets for each section and an activity for each module. Keeping student engagement in mind, we have created a full suite of thought-provoking assignments: discussion prompts, Desmos interactives, application based assignments related to economics, computer science, biology, physics, and engineering, flipped classroom options, and a capstone project.
Faculty can adapt and use these materials in a variety of ways depending on their class structure and institutional requirements. All content is constructed around the goal of helping students master the learning outcomes for this course.
This course is designed to be used as part one of a three-part calculus sequence: Calculus I covers functions, limits, derivatives, and integration, Calculus II covers integration, differential equations, sequences and series, and parametric equations and polar coordinates. Calculus III covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integrations, and second-order differential equations.
Lumen Learning’s Calculus II consists of the following resources:
- Online Homework Manager (OHM): A flexible, user-friendly math homework system with customizable learning content, assessments, and activities you can tailor to fit your needs. Request an OHM instructor account.
- Outcome-aligned OER: Designed to replace expensive textbooks, this course curates open educational resources (OER) aligned with learning outcomes. Teach as-is or customize to fit your needs.
- LMS Integration: This course may be delivered with seamless LMS integration and automatic grade return for Canvas, Blackboard, Brightspace, and Moodle.
- Accessibility: Lumen is 100% committed to providing learning materials that are accessible to all learners. Lumen course materials are mobile-friendly.
Lumen Learning applies learning science research to the design of affordable digital courseware to engage students and empower them in building a strong foundation in Mathematics. OHM has fully digital course materials focused on hands-on practice, customizable template courses, interactive full e-text with immediate feedback that stands alone or integrates seamlessly into Brightspace, Canvas, Blackboard, or Moodle.
Contributors: Lumen Learning vetted and improved sources from Calculus Volume 3 by OpenStax.
Course Content
Module 1: Describing curves through parametric equations and polar coordinates
- Identify parametric equations
- Apply calculus to parametric equations
- Understand polar coordinates and their application
- Determine area and arc length in polar coordinates
- Distinguish properties of parabolas, ellipses, and hyperbolas
Module 2: Interpret functions of two or three independent variables in multidimensional space
- Understand vectors and their operations
- Apply vectors to three-dimensional space
- Use the dot product
- Use the cross product
- Interpret equations of lines and planes in space
- Distinguish properties of cylinders, ellipsoids, paraboloids, and hyperboloids
- Convert coordinates between rectangular and nonrectangular coordinates
Module 3: Interpret vector-valued functions to determine the velocity, acceleration, arc length, and curvature of an object’s trajectory
- Apply vector-valued functions to curves in the plane and in three-dimensional space
- Apply calculus to vector-valued functions
- Determine arc length and curvature in space
- Describe motion in space
Module 4: Develop methods to solve differential equations of functions with several variables
- Interpret functions of several variables
- Determine limits and continuity of functions of several variables
- Calculate the derivatives of functions of several variables
- Apply tangent planes to three-dimensional surfaces
- Apply the chain rule to functions of several variables
- Calculate directional derivatives
- Identify extrema and critical points for a function of two variables
- Apply Lagrange multipliers to solve optimization problems
Module 5: Apply integration techniques to functions containing more than one variable and other coordinate systems
- Calculate double integrals over rectangular regions
- Apply double integrals to general regions
- Use double integrals on polar rectangular regions
- Calculate triple integrals over three-dimensional space
- Evaluate triple integrals using cylindrical and spherical coordinates
- Use triple integrals to locate centers of mass and moments of inertia
- Calculate multiple integrals using a change of variables
Module 6: Generalize vector fields and its application to Green’s, Stokes’ and the divergence theorems
- Identify vector fields
- Calculate line integrals along curves
- Explain conservative vector fields
- Apply Green’s theorem
- Determine divergence and curl for a vector field
- Interpret surface integrals
- Use Stokes’ theorem
- Apply the Divergence theorem
Module 7: Develop methods to solve second-order differential equations. Interpret second-order linear equations
- Solve nonhomogeneous linear equations
- Apply second-order differential equations to real-world concepts
- Use power series to solve differential equations
Teaching with Lumen Course Materials
- Replace expensive textbooks | Ready-to-adopt open educational resources (OER) include text, videos, simulations, self-checks, and other interactives.
- Choose affordability | Low cost to students.
- Use better content | Continuous, data-driven improvements make OER content more effective at supporting learning.
- Simplify access | Easy access to course materials in your LMS (Blackboard, Canvas, D2L, and Moodle) plus automatic grade return.
- Improve student outcomes | Research shows Lumen course materials can improve academic performance, passing rates, and course completion.
FOR INSTRUCTORS
- Connect with students | Instructor and student tools designed to help strengthen communication.
- Save time | Start with curated, outcome-aligned OER and supplemental instructor resources like quiz banks, assignments, slide decks, etc.
- Customize your course | Freedom to tailor course content to fit your learning outcomes and instructional approach.
- Enjoy awesome support | Faculty-friendly onboarding, training, and support.
FOR STUDENTS
- Learn by doing | Online homework, self-check activities, and other interactive tools strengthen learning.
- Engage from day one | Avoid falling behind with access to course materials from the first day of class.
- Retain materials | Download a digital copy of course content to keep forever.
- Become a better learner | Real-time feedback guides students on where to focus and how to improve.
Are you ready to get started with Lumen Learning digital courseware?
Great! Fill out our Explore a Course form, and our team will contact you promptly.
- Zachary Beamer, Piedmont Virginia Community College
- Ryan Melton, Kilgore College
- Matthew Simmons, Florida State College at Jacksonville
- Kaiwen Amrein, Portland Community College
- Weam Al-Tameemi, Texas State University at San Marcos
- William McNeary, Contributor
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- … as well as the eyes and feedback of many other faculty and institutions