# MA103: Multivariable Calculus

Unit 3: Double Integrals and Line Integrals in the Plane   In Single-Variable 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 Single-Variable 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.

In Single-Variable 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 Multi-Variable 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 two-dimensional 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.

This unit will take you approximately 26.75 hours to complete.

☐    Subunit 3.1: 11.5 hours

☐    Subunit 3.1.1: 3.75 hours

☐    Subunit 3.1.2: 1.5 hour

☐    Subunit 3.1.3: 2.75 hours

☐    Subunit 3.1.4: 3.5 hours

☐    Subunit 3.2: 8 hours

☐    Subunit 3.2.1: 3.25 hours

☐    Subunit 3.2.2: 2 hours

☐    Subunit 3.2.3: 2.75 hours

☐    Subunit 3.3: 7.25 hours

☐    Subunit 3.3.1: 1.5 hours

☐    Subunit 3.3.2: 3 hours

☐    Subunit 3.3.3: 1.25 hours

☐    Subunit 3.3.4: 1.5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Evaluate double integrals.
• State the difference between type I and type II integrals.
• Find the volume over type I region.
• Find the volume of solids over type II regions.
• Change iterated integrals from type I to type II and vice versa.
• State Fubini's Theorem, and use it to evaluate integrals.
• Use properties of double integrals to evaluate double integrals.
• Use integrals to calculate the volume and mass.
• Find image of a given region under a given transformation.
• Use a given transformation to evaluate a given iterated integral.
• Construct vector fields.
• Find the gradient vector field.
• Compute divergence and curl of vector fields.
• Evaluate line integrals.
• Derive and apply formulas involving divergence, gradient, and curl.
• Apply Green's Theorem to evaluate line integrals.
• Evaluate Green's Theorem to find the area of a region.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-16-double-integrals/).
``````
• 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 Attribution-NonCommercial-ShareAlike 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 1-4 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.

• 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.

• 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 Attribution-NonCommercial-ShareAlike 3.0 Unported License, they are attributed to Duane Q. Nykamp and the original versions can be found here and here.

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 1-4 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.

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.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-17-polar-coordinates/).
``````
• 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.

• 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.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-18-change-of-variables/).
``````
• 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)

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 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 1-4 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 2-4 at the top of the webpage to continue reading.

• 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.

• 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:

*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.

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](http://mathinsight.org/vector_field_overview)</span>

\*NOTE: In order to view the Java applets within this resource you
must click on the [HTML and
version does not support the Java applets.

<span style="background-color: transparent;"> 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. </span>

This activity should take approximately 30 minutes to complete.

Unported
attributed to Duane Q. Nykamp and the original version can be
found [here](http://mathinsight.org/vector_field_overview).
``````
• 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 Attribution-NonCommercial-ShareAlike 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 1-4 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 2-4 at the top of the webpage.

• 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.

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 1-4 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 2-4 at the top of the webpage to continue reading.

`````` This reading should take approximately 30 minutes to read and
review.

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.

• 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.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-20-path-independence/).
``````
• 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 1-4 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 2-4 at the top of the webpage to continue reading.

• 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.

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.

attributed to Denis Auroux and the original version can be found
``````
• 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 Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-22-greens-theorem/).
``````
• 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 1-4 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 2-4 at the top of the webpage to continue reading.

This reading should take approximately 30 minutes to study.

• 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.

• 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.

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.

attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/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.