# MA103: Multivariable Calculus

Unit 4: Triple Integrals and Surface Integrals in 3-Dimensional 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 3-dimensional region.

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.

This unit will take you approximately 21 hours to complete.

☐    Subunit 4.1: 6 hours

☐    Subunit 4.1.1: 3.5 hours

☐    Subunit 4.1.2: 2.5 hours

☐    Subunit 4.2: 7.25 hours

☐    Subunit 4.2.1: 3.5 hours

☐    Subunit 4.2.2: 3.75 hours

☐    Subunit 4.3: 5.5 hours

☐    Subunit 4.3.1: 1.5 hours

☐    Subunit 4.3.2: 4 hours

☐    Subunit 4.4: 1.5 hours

☐    Subunit 4.5: 0.75 hour

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Find the mass of a solid given the mass density of the solid.
• Calculate area of surfaces.
• Compute the flux of vector fields through a given surface.
• Apply Gauss’ Law.
• Apply Green’s Theorem.
• State and apply the Divergence Theorem.
• Apply Stokes’ Theorem to calculate the work of a vector field around a simple closed curve.
• Apply the Fundamental Theorem for line integrals.
• Find dw for a given differential form w.

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.

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-25-triple-integrals/).
``````
• Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-Space:” Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” (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:” “4.6 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 1-4 of “Section 4.6: Triple Integrals.”  Begin by reading “Part I: Definition of the Triple Integral,” and then 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: “4.6 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.

• 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.

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.

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-26-spherical-coordinates/).
``````
• 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)

This reading should take approximately 30 minutes to study.

• 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.

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.

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-27-vector-fields-in-3d/).
``````
• Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” (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 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 1-4 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.

• 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.

• Activity: The Saylor Foundation: Math Insight’s “Introduction to a Surface Integral of a Scalar-Valued 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 Scalar-Valued 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 Scalar-Valued 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 Attribution-NonCommercial-ShareAlike 3.0 Unported License, they are attributed to Duane Q. Nykamp and the original versions can be found here and here

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.

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-28-divergence-theorem/)
(Lecture 28) and
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-29-divergence-theorem-cont/)
(Lecture 29).
``````
• Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” (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 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:” “5.6 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 1-4 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 2-4 found at the top of the webpage.

This reading should take approximately 30 minutes to study.

• 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.

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.

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-30-line-integrals/).
``````
• Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” (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 sub-subunit 4.3.2.

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.

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.

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-31-stokes-theorem/)
(Lecture 31) and
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-32-stokes-theorem-cont/)
(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 1-4 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.

• 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.

• 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.

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 1-4 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 2-4.

This reading should take approximately 30 minutes to study.

• 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.

4.5 Maxwell’s Equations   - Reading: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” Link: MIT: Denis Auroux’s Multivariable Calculus: “IV. Triple Integrals and Surface Integrals in 3-space” (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 3-space” 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 Attribution-NonCommercial-ShareAlike 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.