# MA103: Multivariable Calculus

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.

This unit will take you approximately 22.25 hours to complete.

☐    Subunit 1.1: 7 hours

☐    Subunit 1.1.1: 2 hours

☐    Subunit 1.1.2: 1 hour

☐    Subunit 1.1.3: 4 hours

☐    Subunit 1.2: 9 hours

☐    Subunit 1.2.1: 2.5 hours

☐    Subunit 1.2.2: 2.5 hours

☐    Subunit 1.2.3: 4 hours

☐    Subunit 1.3: 6.25 hours

☐    Subunit 1.3.1: 1.5 hours

☐    Subunit 1.3.2: 2 hours

☐    Subunit 1.3.3: 1.5 hours

☐    Subunit 1.3.4: 1.25 hour

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define and identify vectors.
• Evaluate the distance between two points in Rand R3.
• Carry out vector operations.
• Represent the operations of vector addition, scalar multiplication, and norm geometrically.
• Determine whether a set in Rn is open, closed, or neither.
• Evaluate a dot product, and interpret it geometrically.
• Evaluate a cross product, and interpret it geometrically.
• Sketch the graph of a plane given its equation.
• Find the equation of the plane.
• Sketch a curve given its parameterization.
• Give the standard parameterization of a circle.
• Compute velocity and acceleration.
• Calculate velocity as a tangent vector.
• Give antiderivatives of vector-valued functions.
• State the properties of the derivative.
• Calculate distance and arc length.
• Compute the unit normal.
• Calculate curvature.

1.1 Vectors   1.1.1 Vectors in R2 and R3   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions:” “Vectors in 2 and 3 Dimensions” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued 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 2-4 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.

• 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.

1.1.2 Vectors in Rn   - Reading: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in Rn” Link: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in Rn (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 Rn” in its entirety (pages 79-87).  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.

displayed on the webpage above.
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• Activity: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in Rn: Exercises with Answers” Link: Brigham Young University: Kenneth Kuttler’s Calculus, Applications and Theory: “Vectors and Points in Rn: 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 88-89.  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).

1.1.3 Inner Product and Cross Product   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued 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: Vector-Valued 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 1-4 in their entirety.

• 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.

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

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.

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-2-determinants/) (Lecture
2) and
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-3-matrices/)
(Lecture 3).
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• 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 Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.

1.2.2 Lines and Planes   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions: 1.4 Lines and Planes” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions: 1.4 Lines and Planes” (HTML and Java)

Instructions: Click on the link above to “Part 1: Equations of Lines,” and read Parts 1-4 in their entirety.  To access Parts 2-4, click on the links to each part at the top of the webpage.

`````` This reading should take approximately 30 minutes to study.

displayed on the webpage above.
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• 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 Attribution-NonCommercial-ShareAlike 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.

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.

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-5-parametric-equations/).
``````
• Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions: 1.5 Parametric Equations” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions: 1.5 Parametric Equations” (HTML and Java)

Instructions: Please read Parts 1-4 in their entirety.  Read the first webpage titled “Part 1: Vector-Valued Functions,” and then click on the links to Parts 2-4 at the top of the page to access each subsequent section.

• 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.

• 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 1-10 under “Examples” at the bottom of the page to access the examples. Each example gives parametric equations and corresponding graphs. Also, try problems 1-6 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.

1.3 Motion   1.3.1 Velocity and Acceleration   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions:” “1.6 Velocity and Acceleration” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued 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 2-4 at the top of the webpage to access the remaining sections of the reading.

`````` This reading should take approximately 30 minutes to complete.

displayed on the webpage above.
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• 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.

1.3.2 Speed and Arc Length   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions: 1.7 Speed and Arc length” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued 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 2-4, 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.

displayed on the webpage above.
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• 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.

• 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.

1.3.3 Components of Acceleration   - Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions:” “1.8 Components of Acceleration” Link: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 1: Vector-Valued Functions:” “1.8 Components of Acceleration” (HTML and Java)

Instructions: Please open the link posted above and read Parts 1-4 in their entirety.  Begin by reading the webpage titled “Part 1: Curvature and the Unit Normal.”  Then, click on the links for Parts 2-4 at the top of the webpage to continue the reading.

`````` This reading should take approximately 30 minutes to study.

displayed on the webpage above.
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• 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.

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.

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-6-keplers-second-law/).
``````
• 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 Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.