Multivariable Calculus
Purpose of Course showclose
Multivariable Calculus is an expansion of SingleVariable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables. The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multivariable functions, in some cases, will yield counterintuitive results.
The structure of this course very much resembles the structure of SingleVariable Calculus I and II. We will begin by taking a fresh look at limits and continuity. With functions of many variables, you can approach a limit from many different directions. We will then move on to derivatives and the process by which we generalize them to higher dimensions. Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals.
The goal of Multivariable Calculus is to provide you with the tools you need to handle problems with several parameters and functions of several variables and to apply your knowledge of their behavior. But a more important goal is to gain a geometrical understanding of what the tools and computations mean.
Course Information showclose
Primary Resources: This course is comprised of the following primary materials:
 East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online
 Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory
 MIT: Denis Auroux’s “Multivariable Calculus” Video Lectures and Lecture Notes
 Unit 1 Assignment: Exercise Sets
 Unit 2 Assignment: Exercise Sets
 Unit 3 Assignment: Exercise Sets
 Unit 4 Assignment: Exercise Sets
 The Final Exam
Time Commitment: This course should take you a total of 88.5 hours to complete. This is only an approximation and may take longer to complete. Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take you about 22 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 7 hours) over three days, for example by completing subsubunit 1.1.1 (a total of 2 hours) on Monday; subsubunit 1.1.2 (a total of 1 hour) and about half of subsubunit 1.1.3 (about 2 hours) on Tuesday; the rest of subsubunit 1.1.3 (about 2 hours) on Wednesday; etc.
Tips/Suggestions: As noted in the “Course Requirements,” SingleVariable Calculus is a prerequisite for this course. If you are struggling with the material as you progress through this course, consider taking a break to revisit MA101 Single Variable Calculus I and MA102 Single Variable Calculus II. It will likely be helpful to have a graphing calculator on hand for this course. If you do not own or have access to one, consider using this free graphing calculator. As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, and definitions that stand out to you. It will be useful to use these notes as a review prior to completing the Final Exam.
Learning Outcomes showclose
 Define and identify vectors.
 Define and compute dot and crossproducts.
 Solve problems involving the geometry of lines, curves, planes, and surfaces in space.
 Define and compute velocity and acceleration in space.
 Define and solve Kepler’s Second Law.
 Define and compute partial derivatives.
 Define and determine tangent planes and level curves.
 Define and compute least squares.
 Define and determine boundaries and infinity.
 Define and determine differentials and the directional derivative.
 Define and compute the gradient and the directional derivative.
 Define, determine, and apply Lagrange multipliers to solve problems.
 Define and compute partial differential equations.
 Define and evaluate double integrals.
 Use rectangular coordinates to solve problems in multivariable calculus.
 Use polar coordinates to solve problems in multivariable calculus.
 Use change of variables to evaluate integrals.
 Define and use vector fields and line integrals to solve problems in multivariable calculus.
 Define and verify conservative fields and path independence.
 Define and determine gradient fields and potential functions.
 Use Green’s Theorem to evaluate and solve problems in multivariable calculus.
 Define flux.
 Define and evaluate triple integrals.
 Define and use rectangular coordinates in space.
 Define and use cylindrical coordinates.
 Define and use spherical coordinates.
 Define and correctly manipulate vector fields in space.
 Evaluate surface integrals and relate them to flux.
 Use the Divergence Theorem (Gauss’ Theorem) to solve problems in multivariable calculus.
 Define and evaluate line integrals in space.
 Apply Stokes’ Theorem to solve problems in multivariable calculus.
 Properly apply Maxwell’s Equations to solve problems.
Course Requirements showclose
√ Have a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plugins or software (e.g., Adobe Reader or Flash).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt., .xls, etc.).
√ Have competency in the English language.
√ Have read the Saylor Student Handbook.
√ Have completed the following courses from “The Core Program” of the mathematics discipline: MA101 SingleVariable Calculus I and MA102 SingleVariable Calculus II.
Preliminary Information

Optional Resource: Monroe Community College's “Exploring Multivariable Calculus” Dynamic Visualization Project
Link: Monroe Community College's “Exploring Multivariable Calculus” (HTML and Java)  Dynamic Visualization Project
Instructions: Please note that this is an optional resource. Click on “Multivariable Calculus Exploration Applet” link on the left side to upload the applet. This is a dynamic visualization tool for multivariable calculus and can be used throughout the course, especially to visualize threedimensional objects.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Unit Outline show close
Expand All Resources Collapse All Resources

Unit 1: Vectors and Matrices
This unit begins with a discussion of vectors and vector algebra. You will study and then be able to define the geometric properties of the dot product and the cross product. You will learn vector algebra, which will allow you to perform operations on vectors themselves and not just their coordinates. This will help you understand the geometric significance of the operations and computations performed. You will then learn about parameterizations of lines and planes using vector algebra and geometry. This way of thinking, i.e. going back and forth between algebra and geometry, will be very important for the rest of the course. Finally, this unit introduces fundamental concepts of velocity and acceleration using vectors; this will allow us to study these concepts in their natural form.
Time Advisory show close
Learning Outcomes show close
 1.1 Vectors

1.1.1 Vectors in R^2 and R^3
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “Vectors in 2 and 3 Dimensions”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “Vectors in 2 and 3 Dimensions” (HTML and Java)
Instructions: Please click on the link above, and read the entire webpage titled “Part 1: Vectors in the Plane.” Once you have finished, click on the links to Parts 24 at the top of each webpage to move on to the subsequent sections. Read all four parts in their entirety. Note: Microsoft Internet Explorer is recommended when viewing this site so that mathematical symbols are correctly displayed.
This reading should take approximately 1 hour to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Vectors in 2 and 3 Dimensions Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Vectors in 2 and 3 Dimensions Exercises” (HTML)
Instructions: On the webpage linked above, click on “Exercises” at the top of the page to link to the problem sets. Please work through exercises 1, 3, 7, 11, 15, 23, and 25. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and then finding section 1.1.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “Vectors in 2 and 3 Dimensions”

1.1.2 Vectors in R^n
 Reading: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in R^n”
Link: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in R^{n}” (PDF)
Instructions: Please scroll down the webpage to the “Lecture Notes and Books” heading, and click on the link titled “Math 214 Notes” to open the PDF. Read “Chapter 4: Vectors and Points in R^{n}” in its entirety (pages 7987). You may want to save this PDF file to your desktop to easily access again in this course.
This reading should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in R^n: Exercises with Answers”
Link: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in R^{n}: Exercises with Answers” (PDF)
Instructions: Please click the above link to access Calculus, Applications and Theory. Complete exercises 1, 2, and 5 from “Section 4.4: Exercises with Answers” on pages 8889. Try to solve each problem independently before checking the answer for each question, which is provided directly below the question.
These exercises should take approximately 30 minutes to complete.
Note: As printed, Problem 1 has some errors that may affect your answer to Problem 2. An updated version of the problem is provided here (PDF).
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in R^n”

1.1.3 Inner Product and Cross Product
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.2 The Inner Product” and “1.3 The Cross Product”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.2 The Inner Product” and “1.3 The Cross Product” (HTML and Java)
Instructions: Please click on both of the links above to read “Section 1.2: The Inner Product” and “Section 1.3: The Cross Product.” For each section, please read Parts 14 in their entirety.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “The Inner Product Exercises” and “The Cross Product Exercises”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “The Inner Product Exercises” and “The Cross Product Exercises” (HTML and Java)
Instructions: Please click the link titled “The Inner Product Exercises” above, and then click on the “Exercises” link at the top of the webpage. Complete exercises 1, 9, 17, 23, and 29. Similarly, click on “The Cross Product Exercises” link above, and then select the “Exercises” link at the top of the webpage. Work through exercises 1, 5, 11, 17, 23, and 25. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and scrolling down to find the corresponding sections (1.2 and 1.3).
These exercises should take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 1: Dot Product”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 1: Dot Product” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please access the link above to view the entire video lecture (38:41). You may also click on the “Transcript” tab on the page to read the lecture.
Viewing this video and pausing to take notes should take approximately 45 minutes to complete.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices: Week 1 Summary”
Link: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices: Week 1 Summary” (PDF)
Instructions: Read the entire PDF document (3 pages). Please note that this reading is paired with the video lecture posted in the subsection above: please watch the lecture before accessing this reading.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.2 The Inner Product” and “1.3 The Cross Product”
 1.2 Matrices and Curves

1.2.1 Matrices
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 2: Determinants” and “Video Lecture 3: Matrices”
Links: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 2: Determinants” and “Video Lecture 3: Matrices” (YouTube)
Also available in:
Adobe Flash, iTunes, or Mp4 (Lecture 2)
Adobe Flash, iTunes, or Mp4 (Lecture 3)
Instructions: Please view the linked lectures in their entirety (Lecture 2: 52:51, Lecture 3: 51:05). You may also click on the “Transcript” tabs on each webpage to read the lectures.
These videos should take approximately 2 hours to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here (Lecture 2) and here (Lecture 3).  Reading: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices:” “Week 2 Summary”
Link: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices: Week 2 Summary” (PDF)
Instructions: Read the entire document (5 pages). Please note that this reading is paired with the video lectures posted in the subsection above, so please read it after watching the videos.
This reading should take approximately 30 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 2: Determinants” and “Video Lecture 3: Matrices”

1.2.2 Lines and Planes
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.4 Lines and Planes”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.4 Lines and Planes” (HTML and Java)
Instructions: Click on the link above to “Part 1: Equations of Lines,” and read Parts 14 in their entirety. To access Parts 24, click on the links to each part at the top of the webpage.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 4: Square Systems”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 4: Square Systems” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the linked lecture (49:02). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch, take notes and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Lines and Planes Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Lines and Planes Exercises” (HTML and Java)
Instructions: On the webpage linked above, click on the “Exercises” link at the top of the page to access the problem sets. Complete exercises 1, 7, 11, 19, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and scrolling down to section 1.4.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.4 Lines and Planes”

1.2.3 Parametric Equations
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 5: Parametric Equations”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 5: Parametric Equations” (YouTube)
Also available in: Adobe Flash, iTunes or Mp4
Instructions: Please view the linked lecture (50:50). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch, take notes and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.5 Parametric Equations”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.5 Parametric Equations” (HTML and Java)
Instructions: Please read Parts 14 in their entirety. Read the first webpage titled “Part 1: VectorValued Functions,” and then click on the links to Parts 24 at the top of the page to access each subsequent section.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Parametric Equations Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Parametric Equations Exercises” (HTML and Java)
Instructions: On the webpage linked above, click on the “Exercises” link at the top of the page to access the question sets. Work on exercises 1, 5, 11, 17, 25, and 29. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.5.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: University of Rhode Island: Barbara Kaskosz’s Flash Mathlets: "Parametric Curves" and “Parametric Surfaces”
Link: University of Rhode Island: Barbara Kaskosz' Flash Mathlets: “Parametric Curves” (Flash Activity) and “Parametric Surfaces” (Flash Activity)
Instructions: After opening the webpage “Parametric Curves” linked above, click on links 110 under “Examples” at the bottom of the page to access the examples. Each example gives parametric equations and corresponding graphs. Also, try problems 16 listed under “Problems” to see if you can draw out the graph from given equations. After working out each problem, click on the graph to see the corresponding curve for the equations. Similarly, click on the above link titled “Parametric Surfaces” and work through examples and problems. Feel free to create and graph your own parametric equations to see what the curves look like.
These activities should take approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 5: Parametric Equations”
 1.3 Motion

1.3.1 Velocity and Acceleration
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.6 Velocity and Acceleration”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.6 Velocity and Acceleration” (HTML and Java)
Instructions: Please access the webpage above and read all four sections in their entirety. Begin by reading the first webpage linked above titled “Part 1: Limits and Derivatives.” Then, click on the links to Parts 24 at the top of the webpage to access the remaining sections of the reading.
This reading should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Velocity and Acceleration Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Velocity and Acceleration Exercises” (HTML and Java)
Instructions: Open the webpage linked above and please select the “Exercises” link at the top of the page to access the questions. Complete exercises 1, 5, 9, 13, 25, and 29. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.6.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.6 Velocity and Acceleration”

1.3.2 Speed and Arc Length
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.7 Speed and Arc length”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.7 Speed and Arc length” (HTML and Java)
Instructions: Please read all four sections in their entirety. Begin by reading the first webpage titled “Part 1: Properties of the Derivative.” To access the remaining Parts 24, make sure to click on the link for each part at the top of the webpage.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Speed and Arc length Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Speed and Arc length Exercises” (HTML)
Instructions: On the webpage linked above, click on the “Exercises” link at the top of the webpage to access the questions. Work through exercises 3, 5, 9, 13, 21, and 27. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.7.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “The Arc Length of a Parametrized Curve”
Link: The Saylor Foundation: Math Insight’s “The Arc Length of a Parametrized Curve” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to go through more examples by clicking on the link on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions: 1.7 Speed and Arc length”

1.3.3 Components of Acceleration
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.8 Components of Acceleration”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.8 Components of Acceleration” (HTML and Java)
Instructions: Please open the link posted above and read Parts 14 in their entirety. Begin by reading the webpage titled “Part 1: Curvature and the Unit Normal.” Then, click on the links for Parts 24 at the top of the webpage to continue the reading.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Components of Acceleration Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Components of Acceleration Exercises” (HTML and Java)
Instructions: Please open the webpage linked above and click on the “Exercises” link at the top of the page to access the questions. Complete exercises 3, 5, 9, 11, and 17. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.8.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: VectorValued Functions:” “1.8 Components of Acceleration”

1.3.4 Kepler’s Second Law
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 6: Kepler's Second Law”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 6: Kepler's Second Law” (Adobe Flash, iTunes, or Mp4)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the linked lecture in its entirety (48:04). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices”
Link: MIT: Denis Auroux’s Multivariable Calculus: “I. Vectors and Matrices” (PDF)
Instructions: Read the entire PDF file (3 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 6: Kepler's Second Law”

Unit 2: Partial Differentiation
For singlevariable functions, the derivative of the graph at a specified point is the slope of the tangent line at that point. This is because the dependent variable is only changing with regards to a single independent variable. In this course, we will consider functions of multiple variables. Because these functions have outputs that depend on multiple variables, each variable contributes to the rate of change of the function independently. Thus, we refer to the function’s rate of change with respect to a single specified variable as a partial derivative, as it does not describe the entire rate of change of the function. Moreover, for functions of multiple variables, there will no longer be a single tangent line at a specified point. For example, in the case of a function of two variables, we consider a tangent plane as opposed to a tangent line. The concept of a partial derivative is important to study concepts such as tangent spaces, extrema, etc. in the context of functions of 2 or more variables.
Time Advisory show close
As in SingleVariable Calculus, Multivariable Calculus studies the maximum and minimum values for given functions due to their numerous applications. Lagrange Multipliers is a method of finding maximum and minimum values for functions subject to constraints that uses partial derivatives; however, the idea of optimization with constraints has no onevariable analogue.
Again, you should focus on the geometric meaning of the concepts introduced in this unit.
Learning Outcomes show close
 2.1 Partial Derivatives

2.1.1 Level Curves and Partial Derivatives
 Activity: The Saylor Foundation: Math Insight's “Level Sets”
Link: The Saylor Foundation: Math Insight’s “Level Sets” (PDF)
Also Available in:
HTML and Java
NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Please click on the webpage linked above and work through the notes and the applets. Feel free to go through more examples by clicking on the link at the bottom of the page.
This activity should take approximately 1 hour to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.  Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 8: Partial Derivatives”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 8: Partial Derivatives” (Adobe Flash, iTunes or Mp4)
Also available in: Adobe Flash, iTunes or Mp4
Instructions: Please view the entire video lecture (46:13). You may also click on the “Transcript” tab on the page to read the lecture.
This reading should take approximately 15 minutes to read and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives:”
Link: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives” (PDF)
Instructions: Read the entire PDF document (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This activity should take approximately 15 minutes to complete.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Activity: The Saylor Foundation: Math Insight's “Level Sets”

2.1.2 MaxMin Problems and Least Squares
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 9: MaxMin and Least Squares”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 9: MaxMin and Least Squares” (Youtube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire lecture (49:44). You may also click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 9: MaxMin and Least Squares”

2.1.3 Second Derivative Test
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 10: Second Derivative Test”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 10: Second Derivative Test” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire lecture (52:18). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 10: Second Derivative Test”

2.1.4 Limits, Continuity, and Partial Differentiation
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.1 Functions of 2 Variables,” “2.2 Limits and Continuity,” and “2.3 Partial Derivatives”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.1 Functions of 2 Variables,” (HTML and Java) “2.2 Limits and Continuity,” (HTML) and “2.3 Partial Derivatives” (HTML and Java)
Instructions: Please click on each link above to sections “2.1 Functions of 2 Variables,” “2.2 Limits and Continuity,” and “2.3 Partial Derivatives.” Read all four parts for each individual section. Begin by reading the first webpage for each section linked above, and then select the other Parts at the top of each webpage to continue reading (After finishing Part 4 of "2.3 Partial Derivatives", you will have read 12 Parts total).
This reading should take approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Functions of 2 Variables Exercises,” “Limits and Continuity Exercises,” and “Partial Derivatives Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Functions of 2 Variables Exercises,” (HTML and Java) “Limits and Continuity Exercises,” (HTML) and “Partial Derivatives Exercises” (HTML and Java)
Instructions: Please click on the “Functions of 2 Variables Exercises” link above, and at the top of this webpage, click on the “Exercises” link to access the question sets. Work through exercises 7, 9, 15, 19, and 25. Similarly, click on the “Limits and Continuity Exercises” link above, and once on the webpage, select the “Exercises” link at the top of the webpage to access the question sets. Complete exercises 5, 13, 17, and 25. Finally, click on the “Partial Derivatives Exercises” link above, and at the top of this webpage, click on the “Exercises” link to be redirected to the question sets. Try to solve exercises 5, 11, 19, 25, and 29. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and then finding sections 2.12.3.
These exercises should take approximately 3 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight's "Introduction to Partial Derivatives"
Link: The Saylor Foundation: Math Insight’s “Introduction to Partial Derivatives” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.1 Functions of 2 Variables,” “2.2 Limits and Continuity,” and “2.3 Partial Derivatives”
 2.2 Differentiation and Chain Rule

2.2.1 Chain Rule
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 11: Chain Rule”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 11: Chain Rule” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire lecture (50:09). You may also click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives”
Link: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives” (PDF)
Instructions: Read the entire PDF file (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.5 Linearization and the Hessian” and “2.6 The Chain Rule”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.5 Linearization and the Hessian” (HTML and Java) and “2.6 The Chain Rule” (HTML)
Instructions: Please click the links titled “Section 2.5 Linearization and the Hessian” and “Section 2.6 The Chain Rule” above. For each section, read Parts 14 in their entirety. You may access each part by clicking on the links at the top of each webpage.
These exercises should take approximately 3 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.5 Linearization and the Hessian” and “2.6 The Chain Rule”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.5 Linearization and the Hessian” (HTML) and “2.6 The Chain Rule” (HTML)
Instructions: Please click the link titled “2.5 Linearization and the Hessian” posted above, and then select the “Exercises” link at the top of the webpage to access the questions. Try to solve exercises 1, 9, 21, and 27. Then, click on the “2.6 The Chain Rule” link posted above, and select the “Exercises” link at the upper right corner of the webpage to access the questions. Complete exercises 5, 13, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and then scrolling down to find sections 2.5 and 2.6.
These exercises should take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 11: Chain Rule”

2.2.2 Gradient
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 12: Gradient”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 12: Gradient” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view entire video lecture (50:10). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.7 Properties of the Gradient”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.7 Properties of the Gradients” (HTML and Java)
Instructions: Please read all four parts of “2.7 Properties of the Gradients.” Begin by reading “Part 1: Gradients and Level Curves” linked above, and then select the links for Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.7 Properties of the Gradients Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.7 Properties of the Gradients Exercises” (HTML)
Instructions: Please click on the link above titled “2.7 Properties of the Gradients Exercises.” On this webpage, click on the link titled “Exercises” at the upper right corner of the webpage to access the questions. Try to complete exercises 3, 13, and 15. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.7.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight's “An Introduction to the Directional Derivative and the Gradient”
Link: The Saylor Foundation: Math Insight’s “An Introduction to the Directional Derivative and the Gradient” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 12: Gradient”

2.2.3 Optimization and Lagrange Multipliers
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 13: Lagrange Multipliers”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 13: Lagrange Multipliers” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (50:10). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.8 Optimization” and “2.9 Lagrange Multipliers”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.8 Optimization” (HTML and Java) and “2.9 Lagrange Multipliers” (HTML and Java)
Instructions: Please click the links titled “Section 2.8 Optimization” and “Section 2.9 Lagrange Multipliers” found above. Read Parts 14 of each section. Make sure to click on the links to each part at the top of each webpage to complete this reading assignment.
This reading should take approximately 1 hour to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.8 Optimization Exercises” and “2.9 Lagrange Multipliers Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.8 Optimization Exercises” (HTML and Java) and “2.9 Lagrange Multipliers Exercises” (HTML and Java)
Instructions: Please click on the link above titled “2.8 Optimization Exercises”, and then select the “Exercises” link at the top right corner of the webpage to access the question sets. Solve exercises 1, 17, and 25. Similarly, click on the link titled “2.9 Lagrange Multipliers Exercises” above, and then click on the link titled “Exercises” at the upper right corner of the webpage to access the questions. Work on exercises 5, 7, and 15. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to sections 2.8 and 2.9.
These exercises should take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: University of Minnesota: Jonathan Rogness’ Multivariable Calculus and Vector Analysis: “Lagrange Multipliers”
Link: University of Minnesota: Jonathan Rogness’ Multivariable Calculus and Vector Analysis: “Lagrange Multipliers” (HTML and Java)
Instructions: Click on the webpage linked above and work through the notes and the applets.
This activity should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 13: Lagrange Multipliers”

2.2.4 Partial Differential Equations
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 14: NonIndependent Variables” and “Video Lecture 15: Partial Differential Equations”
Links: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 14: NonIndependent Variables” (YouTube) and “Video Lecture 15: Partial Differential Equations” (YouTube)
Also available in:
Adobe Flash, iTunes, or Mp4 (Lecture 14)
Adobe Flash, iTunes, or Mp4 (Lecture 15)
Instructions: Please view the video lectures in their entirety (49:11 and 45:23, respectively). You may also click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here (Lecture 14) and here (Lecture 15).  Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives:”
Link: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives” (PDF)
Instructions: Open the link above and read the entire PDF document (3 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.4: Partial Differential Equations”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.4: Partial Differential Equations” (HTML)
Instructions: Please read Parts 14 of “Section 2.4: Partial Differential Equations” from the link posted above. Begin by reading “Part 1: Partial Differential Equations” linked above, and then click on the links to Parts 24 at the top of the webpage.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.4: Partial Differential Equations Exercises”
Link: East Tennessee State University’s Jeff Knisley’s Multivariable Calculus Online: “2.4: Partial Differential Equations Exercises” (HTML)
Instructions: Please click on the “2.4: Partial Differential Equations Exercises” link above, and once on the webpage, click on the “Exercises” link at the top right corner of the webpage to redirect to the exercise sets. Complete exercises 11, 21, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.4.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 14: NonIndependent Variables” and “Video Lecture 15: Partial Differential Equations”

Unit 3: Double Integrals and Line Integrals in the Plane
In SingleVariable Calculus, you learned that the integral of a function is the area below the graph of the function and over a specified interval. The double integral of a function of two variables is the volume below the graph of the function and over a specified region. In SingleVariable Calculus, you approximated the area under a curve by taking slices of the area. You will now approximate the volume under a function by taking slices of the entire volume.
Time Advisory show close
In SingleVariable Calculus, you learned about results such as the area of a region, volume of a solid, and length of a curve using definite integrals. In this unit of MultiVariable Calculus, we will develop the theory of multiple integrals to determine similar results.
You will also learn about Green’s Theorem; it defines the relationship between line integrals and double integrals, allowing us to reduce possibly complicated line integrals to a potentially simpler double integral. Please note that Green's Theorem is a twodimensional case of the more general Stokes’ Theorem, which we will discuss in the next unit.
Finally, you will learn about flux; it is a scalar quantity important to the fields of mathematics and physics. It is derived from the surface integral over a specified region of a particular vector field. Using what we know about fields and integrals, we can look at the physical interpretations of flux.
Learning Outcomes show close
 3.1 Multiple Integrals

3.1.1 Double Integrals
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 16: Double Integrals”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 16: Double Integrals” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (48:00). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane”
Link: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane” (PDF)
Instructions: Read the entire PDF document (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals”, “4.1 Iterated Integrals” and “4.2 The Double Integral”
Links: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.1 Iterated Integrals” (HTML and Java) and “4.2 The Double Integral” (HTML)
Instructions: Please open and read Parts 14 of both “Section 4.1 Iterated Integrals” and “Section 4.2 The Double Integral.” To access each section, click on the links to each part at the top of the webpages.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.1 Iterated Integrals Exercises” and “4.2 The Double Integral Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.1 Iterated Integrals Exercises” (HTML) and “4.2 The Double Integral Exercises” (HTML)
Instructions: Please click on the “4.1 Iterated Integrals Exercises” link above, and then select the “Exercises’ link at the top right hand of the webpage to access the questions. Work through exercises 7, 13, 21, and 23. Once you have completed those exercises, click the link titled “4.2: Double Integrals Exercises” above, and select the “Exercises” link at the top of the webpage to redirect to the question sets. Try to complete exercises 7, 13, 21, and 29. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate sections 4.1 and 4.2.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “Introduction to Double Integrals” and “Double Integrals as Volume”
Link: The Saylor Foundation: Math Insight’s “Introduction to Double Integrals” (PDF) and “Double Integrals as Volume” (PDF)
Also Available in:
HTML and Java (Introduction to Double Integrals)
HTML and Java (Double Integrals as Volume)
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java links, as the PDF version does not support the Java applets.
Instructions: Click on the webpages linked above and work through the notes and the applets. Feel free to work on more examples or reading more sections by clicking the relevant links on the bottom of the pages.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resources above are released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, they are attributed to Duane Q. Nykamp and the original versions can be found here and here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 16: Double Integrals”

3.1.2 Applications of the Double Integral
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.3 Applications of the Double Integral”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.3 Applications of the Double Integral” (HTML and Java)
Instructions: Please read Parts 14 of “Applications of the Double Integral” from the link posted above. Begin by reading the webpage titled “Part 1: Mass Density.” Then, click on the link to each additional part at the top of the webpage.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.3 Applications of the Double Integral Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.3 Applications of the Double Integral Exercises” (HTML and Java)
Instructions: Please click on the link titled “4.3 Applications of the Double Integral Exercises” above, and once on the webpage, select the “Exercises” link on the upper right corner of the webpage to access the questions. Work through exercises 1, 11, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate section 4.3.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.3 Applications of the Double Integral”

3.1.3 Double Integrals in Polar Coordinates
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 17: Polar Coordinates”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 17: Polar Coordinates” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (51:30). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.5 Integration in Polar Coordinates”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals: 4.5 Integration in Polar Coordinates” (HTML)
Instructions: Please click on the link above titled “4.5 Integration in Polar Coordinates.” Begin by reading the webpage for “Part 1: Change of Variable into Polar Coordinates.” Then, click on the link to each additional part at the top of the webpage; read all four parts in their entirety.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.5 Integration in Polar Coordinates Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.5 Integration in Polar Coordinates Exercises” (HTML)
Instructions: Please click the link titled “Section 4.5 Double Integrals in Polar Coordinates Exercises” found above, and once on the webpage, click on the link titled “Exercises” at the upper right corner of the webpage to access the questions. Solve exercises 5, 11, 21, and 27. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate section 4.5.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 17: Polar Coordinates”

3.1.4 Change of Variables
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 18: Change of Variables”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 18: Change of Variables” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (49:55). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane”
Link: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane” (PDF)
Instructions: Read the entire document (5 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to read and review.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals: 4.4 Change of Variable”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals: 4.4 Change of Variable” (HTML)
Instructions: Please read Parts 14 of “Section 4.4 Change of Variable.” Begin by reading “Part 1: Area of the Image of a Region” which the link above will take you to, after you complete Part 1, click on the links to Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.4 Change of Variable Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.4 Change of Variable Exercises” (HTML)
Instructions: Please click the link titled “4.4: Change of Variable Exercises” above, and once on the webpage, select the “Exercises” link to redirect to the question sets. Work on exercises 5, 15, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate section 4.4.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “Introduction to Changing Variables in Double Integrals”
Link: The Saylor Foundation: Math Insight’s “Introduction to Changing Variables in Double Integrals” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 18: Change of Variables”
 3.2 Vector Fields and Line Integrals

3.2.1 Vector Fields
 Activity: The Saylor Foundation: Math Insight’s “Vector Field Overview”
Link: The Saylor Foundation: Math Insight’s “Vector Field Overview” (HTML and Java)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.  Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 19: Vector Fields”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 19: Vector Fields” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (51:09). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.1 Vector Fields”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.1 Vector Fields” (HTML and Java)
Instructions: Please read Parts 14 of “Section 5.1: Vector Fields.” Begin by reading “Part 1: Vector Fields” found above, and then continue on by selecting the links to Parts 24 at the top of the webpage.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.1 Vector Fields Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.1 Vector Fields Exercises” (HTML and Java)
Instructions: Please click on the “5.1 Vector Fields Exercises” link above, and once on the webpage, click on the “Exercises” link at the top of the webpage to redirect to the exercise sets. Work on questions 1, 5, 17, 21, and 25. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 5, then locate section 5.1.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Activity: The Saylor Foundation: Math Insight’s “Vector Field Overview”

3.2.2 Line Integrals
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.2 Line Integrals”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.2 Line Integrals” (HTML and Java)
Instructions: Read Parts 14 of “Section 5.2: Line Integrals.” Begin by reading the first webpage which can be found in the link above (“Part 1: Line Integrals over Parameterized Curves”), and then select the links to Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.2 Line Integrals Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.2 Line Integrals Exercises” (HTML)
Instructions: Please click the link titled “Section 5.2: Line Integrals” above, and then select the “Exercises” link at the upper right hand corner of the webpage to access the question sets. Try to solve exercises 5, 9, 21, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 5, then locate section 5.2.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “An Introduction to a Line Integral of a Vector Field”
Link: The Saylor Foundation: Math Insight’s “An Introduction to a Line Integral of a Vector Field” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.2 Line Integrals”

3.2.3 Path Independence and Conservative Fields
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 20: Path Independence”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 20: Path Independence” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (50:23). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.3 Potentials of Conservative Fields”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.3 Potentials of Conservative Fields” (HTML and Java)
Instructions: Please read Parts 14 of Section 5.3. Begin by reading “Part 1: Finding Potentials” which can be found through the link above, and then select the links to Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.3 Potentials of Conservative Fields Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.3 Potentials of Conservative Fields Exercises” (HTML)
Instructions: Please click the link titled “Section 5.3: Potentials of Conservative Fields” above, and then, click on the link titled “Exercises” at the upper right corner of the webpage to redirect to the questions. Try to solve exercises 7, 13, 21, and 27. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 5, then locate section 5.3.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 20: Path Independence”
 3.3 Fundamental Theorems

3.3.1 Gradient Fields
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 21: Gradient Fields”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 21: Gradient Fields” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (50:11). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane”
Link: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane” (PDF)
Instructions: Read the entire PDF document (5 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 21: Gradient Fields”

3.3.2 Green’s Theorem
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 22: Green's Theorem”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 22: Green's Theorem” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (46:45). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.4 Green's Theorem”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.4 Green's Theorem” (HTML)
Instructions: Please read Parts 14 of “Section 5.4: Green’s Theorem.” Begin by reading “Part 1: Double Integrals and Boundary Curves” which is linked above, and then select the links to Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.4 Green’s Theorem Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.4 Green's Theorem Exercises” (HTML)
Instructions: Please click the link titled “Section 5.4: Green's Theorem” above, and then select the link titled “Exercises” at the upper right corner of the webpage to access the questions. Complete exercises 1, 5, 19, 23, and 27. You may check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 5, then locate section 5.4.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “The Idea Behind Green’s Theorem”
Link: The Saylor Foundation: Math Insight’s “The Idea Behind Green’s Theorem” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 22: Green's Theorem”

3.3.3 Flux
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 23: Flux”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 23: Flux” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (50:13). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 23: Flux”

3.3.4 Simply Connected Region
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 24: Simply Connected Regions”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 24: Simply Connected Regions” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (49:00). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane”
Link: MIT: Denis Auroux’s Multivariable Calculus: “III. Double Integrals and Line Integrals in the Plane” (PDF)
Instructions: Read the entire PDF document (4 pages) found in the link above. Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 24: Simply Connected Regions”

Unit 4: Triple Integrals and Surface Integrals in 3Dimensional Space
One common misconception of multiple integrals is that the primary difference is the units of the output of our integration. When we used single integrals, our output was area; when we worked with double integrals, our output was volume. It may be natural to assume that triple integrals give an output of whatever comes next, but this is not the case. The true variation between multiple integrals is not the output, or even the function, but the domain of integration. Single integrals are evaluated over an interval, whereas double integrals are evaluated over a planar region. Triple integrals are similar to other integrals in most regards, but they are evaluated over a 3dimensional region.
Time Advisory show close
After learning about the basics of triple integrals, you will learn about The Divergence Theorem. As with Green's Theorem, the Divergence Theorem is a special case of the more generalized Stokes' Theorem that we will see later in this unit. The Divergence Theorem states that the flux of a vector field is equivalent to the volume integral of the divergence of the specified region. Essentially, this proves that the net flow leaving our region is equal to the total amount of sources (minus sinks) in our region. This brings us to Stokes’ Theorem, which generalizes one of the most important theorems in Calculus, the Fundamental Theorem of Calculus. Stokes’ Theorem defines a relationship between the integration of various differential forms over manifolds in multiple dimensions.
Finally, you will learn a little about Maxwell's Equations. They are a set of four different partial differential equations: Gauss' Law, Gauss' Law for Magnetism, Faraday's Law, and Ampère's Law with Maxwell's Correction. Each equation defines a relationship between electric fields, magnetic fields, density, and various other important mathematical parameters.
Learning Outcomes show close
 4.1 Triple Integrals

4.1.1 Introduction to Triple Integrals
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 25: Triple Integrals”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 25: Triple Integrals” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (48:42). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3Space:”
Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space” (PDF)
Instructions: Please access the link above and read the entire PDF document (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to read and review.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.6 Triple Integrals”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals: 4.6 Triple Integrals” (HTML and Java)
Instructions: Read Parts 14 of “Section 4.6: Triple Integrals.” Begin by reading “Part I: Definition of the Triple Integral,” and then click on the links to Parts 24 at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.6 Triple Integrals Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.6 Triple Integrals Exercises” (HTML and Java)
Instructions: Please click on the link above, and once on the webpage, click the link titled “Exercises” at the upper right corner of the webpage to access the questions. Work through exercises 5, 13, and 21. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate section 4.6.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “Introduction to Triple Integrals”
Link: The Saylor Foundation: Math Insight’s “Introduction to Triple Integrals” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 25: Triple Integrals”

4.1.2 Cylindrical and Spherical Coordinates
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 26: Spherical Coordinates”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 26: Spherical Coordinates” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (51:05). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals:” “4.7 Spherical Coordinates”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 4: Multiple Integrals: 4.7 Spherical Coordinates” (HTML and Java)
Instructions: Please read Parts 14 of “Section 4.7: Spherical Coordinates.” Begin by reading “Part I: Triple Integrals in Cylindrical Coordinates” linked above. Once you are finished reading Part 1, select the links to each subsequent part at the top of the webpage until you have completed reading all four webpages. Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.7 Spherical Coordinates Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “4.7 Spherical Coordinates Exercises” (HTML)
Instructions: Please click on the link above titled “Section 4.7: Spherical Coordinates,” and then click on the “Exercises” link at the upper right corner of the webpage to redirect to the question sets. Try to solve the following exercises: 5, 13, 21, and 27. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 4, then locate section 4.7.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 26: Spherical Coordinates”
 4.2 The Divergence Theorem

4.2.1 Vector Fields in 3D and Surface Integrals
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 27: Vector Fields in 3D”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 27: Vector Fields in 3D” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (50:34). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space”
Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space” (PDF)
Instructions: Access the link above and read the entire PDF document (5 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.
This reading should take approximately 15 minutes to read and review.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.5 Surface Integrals”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.5 Surface Integrals” (HTML and Java)
Instructions: Please read Parts 14 of “Section 5.5: Surface Integrals.” Begin by reading the first webpage linked above, and then select the link to each part at the top of the webpage to continue reading.
This reading should take approximately 30 minutes to read and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.5 Surface Integrals Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.5 Surface Integrals Exercises” (HTML and Java)
Instructions: Please click on the “5.5 Surface Integrals” link above, and then select the “Exercises” link at the upper right corner of the webpage to redirect to the questions. Complete exercises 3, 5, 17, and 25. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF); on the main page, choose chapter 5, then locate section 5.5.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “Introduction to a Surface Integral of a ScalarValued Function” and “Introduction to a Surface Integral of a Vector Field”
Link: The Saylor Foundation: Math Insight’s “Introduction to a Surface Integral of a ScalarValued Function” (PDF) and “Introduction to a Surface Integral of a Vector Field” (PDF)
Also Available in:
HTML and Java (Introduction to a Surface Integral of a ScalarValued Function)
HTML and Java (Introduction to a Surface Integral of a Vector Field)
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java links, as the PDF version does not support the Java applets.
Instructions: Click on the webpages linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete
Terms of Use: The linked resources above are released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, they are attributed to Duane Q. Nykamp and the original versions can be found here and here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 27: Vector Fields in 3D”

4.2.2 The Divergence Theorem
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 28: Divergence Theorem” and “Video Lecture 29: Divergence Theorem (cont.)”
Links: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 28: Divergence Theorem” (YouTube) and “Video Lecture 29: Divergence Theorem (cont.)” (YouTube)
Also available in:
Adobe Flash, iTunes, or Mp4 (Lecture 28)
Adobe Flash, iTunes, or Mp4 (Lecture 29)
Instructions: Please view both video lectures in their entirety (49:16 and 50:13, respectively). You may also click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here (Lecture 28) and here (Lecture 29).  Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space”
Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space” (PDF)
Instructions: Open the link posted above and read the entire PDF document (2 pages). Please note that this reading is paired with the video lecture from the subsection above so please read it after watching the video.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.6 The Divergence Theorem”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.6 The Divergence Theorem” (HTML and Java)
Instructions: Please read Parts 14 of “Section 5.6: The Divergence Theorem.” Begin by reading “Part I: The Divergence Theorem” which is linked above, and then continue on by selecting the links to Parts 24 found at the top of the webpage.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.6 The Divergence Theorem Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.6 The Divergence Theorem Exercises” (HTML and Java)
Instructions: Please click on the link above titled “5.6 The Divergence Theorem Exercises.” Once on the webpage, select “Exercises” at the upper right corner of the webpage to access the questions. Try to complete exercises 1, 7, 11, and 19. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF). On the main page, choose chapter 5, then locate section 5.6.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 28: Divergence Theorem” and “Video Lecture 29: Divergence Theorem (cont.)”
 4.3 Stokes’ Theorem

4.3.1 Line Integrals
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 30: Line Integrals”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 30: Line Integrals” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: Please view the entire video lecture (49:42). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 1 hour and 15 minutes to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space”
Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space” (PDF)
Instructions: Open and read the entire PDF document (5 pages) posted above. This reading is paired with the video lecture above so please read it after watching the video. Please note that this resource also covers the topic outlined in subsubunit 4.3.2.
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 30: Line Integrals”

4.3.2 Stokes’ Theorem
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 31: Stokes' Theorem” and “Video Lecture 32: Stokes' Theorem (cont.)”
Links: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 31: Stokes' Theorem” (YouTube) and “Video Lecture 32: Stokes' Theorem (cont.)” (YouTube)
Also available in:
Adobe Flash, iTunes, or Mp4 (Lecture 31)
Adobe Flash, iTunes, or Mp4 (Lecture 32)
Instructions: Please view both video lectures in their entirety (48:21 and 50:09, respectively). You may also click on the “Transcript” tab on the page to read the lecture.
These videos should take approximately 2 hours to watch and review.
Terms of Use: The video above is released under Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here (Lecture 31) and here (Lecture 32).  Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems:” “5.7 Stokes’ Theorem”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.7 Stokes’ Theorem” (HTML and Java)
Instructions: Please read Parts 14 of “Section 5.7: Stokes’ Theorem.” Begin by reading “Part I: Stokes’ Theorem” which is linked above, and then select the links to the remaining parts at the top of the webpage.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.7 Stokes’ Theorem Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.7 Stokes’ Theorem Exercises” (HTML and Java)
Instructions: Please click on the link above titled “5.7 Stokes’ Theorem Exercises,” and then select the “Exercises” link at the upper right corner of the webpage to access the questions. Try to solve exercises 1, 7, 11, 17 and 19. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page; choose chapter 4, then locate section 4.6.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: The Saylor Foundation: Math Insight’s “The Idea behind Stoke’s Theorem”
Link: The Saylor Foundation: Math Insight’s “The Idea behind Stoke's Theorem” (PDF)
Also Available in:
HTML and Java
*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.
Instructions: Click on the webpage linked above, and work through the notes and the applets. Feel free to work on more examples, or read more sections by clicking the relevant links on the bottom of the page.
This activity should take approximately 30 minutes to complete.
Terms of Use: The linked resource above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.
 Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 31: Stokes' Theorem” and “Video Lecture 32: Stokes' Theorem (cont.)”

4.4 Differential Forms
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.8 Differential Forms”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.8 Differential Forms” (HTML and Java)
Instructions: This reading is optional. Please read Parts 14 of “Section 5.8: Differential Forms.” Begin by reading the first webpage linked above, and then select the links at the top of the webpage to Parts 24.
This reading should take approximately 30 minutes to study.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.8 Differential Forms Exercises”
Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “5.8 Differential Forms Exercises” (HTML)
Instructions: This assessment is optional. Please click on the “5.8 Differential Forms Exercises” link above, and once the webpage has opened, select the “Exercises” link at the top of the webpage to access the questions. Complete exercises 5, 7, 15, 17, and 25. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choose chapter 5, then locate section 5.8.
These exercises should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 5: Fundamental Theorems: 5.8 Differential Forms”

4.5 Maxwell’s Equations
 Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space”
Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space” (PDF)
Instructions: This reading is optional. Please click on the PDF link after “Week 14 Summary” under the “IV. Triple Integrals and Surface Integrals in 3space” heading. Read the entire PDF document (2 pages).
This reading should take approximately 15 minutes to study.
Terms of Use: The article above is released under a Creative Commons AttributionNonCommercialShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.  Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 33: Maxwell's Equations”
Link: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 33: Maxwell's Equations” (YouTube)
Also available in: Adobe Flash, iTunes, or Mp4
Instructions: This lecture is optional. Please click on the link above, and view the entire video lecture (28:23). You may also click on the “Transcript” tab on the page to read the lecture.
This video should take approximately 30 minutes to watch and review.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3space”

Final Exam
 Final Exam: The Saylor Foundation's MA103 Final Exam
Link: The Saylor Foundation's MA103 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation's MA103 Final Exam