# Calc III

### Spring 2018

Beauty is the first test: there is no permanent place in the world for ugly mathematics” G. H. Hardy

To generate other beautiful symmetric parametrized curves in Mathematica, please refer to the symmetric curves

Handouts and Misc

0. Trig Review

Parametrization

Polar coordinates

Area in polar coordinates

Vectors in 2D and 3D

Dot and Cross products (lecture conducted in class)

Lines and Planes (partially conducted in class)

Cylindrical and Spherical Coordinates

Vector-valued functions

### Fall 17

Blackboard assignments

1. Trig review: due on September 14, 2017 11:59:00 PM EDT
2. Parametrization: due on September 17, 2017 11:59:00 PM EDT. Homework cannot be started past this date

### Week 1

In the first week, we went over the syllabus and over important rules for the course. By the end of this week, you should have watched all 8 videos below

Furthermore, you should read the following review guide: Trig Review guide. Next, you should complete Set 0   (Set 0 Solutions). Once you your are done reviewing your Trigonometry, you may move on to the following list.

1. Complete the Trig review and submit your answers on Blackboard. You are allowed 2 attempts. You must submit your solutions by September 14, 2017, 11:59:00 PM EDT.
2. Read Section 11.1 (parametric curves, see Reading-assignments-sections-11-1-11-3-11-4)
3. Homework Set 1

### Week 2

A plane curve is determined by a pair of parametric equations x=f(t) and y=g(t) where t belongs to some interval I. The pair of relationship x=f(t) and y=g(t) together with the interval I is called a parametrization of the curve.

Examples

• A line segment starting at (a,b) and ending at (c,d) may be parametrized as follows: $x=a+t(c-a), y=b+t(d-b)$ and $t\in[0,1].$

•  A circle of radius R with center (a,b) has for parametrization $x=a+R \cos(\theta),y=b+R \sin(\theta)$ for θ∈[0,2π) or θ∈[-π,π)
• An ellipse with equation $\left( x/a\right) ^{2}+\left( y/b\right) ^{2}=1, \text{with} a,b>0$ is parametrized as follows $x=a \cos(\theta),y=b \sin(\theta)$ for θ∈[0,2π) or θ∈[-π,π)
• A cycloid is a curve traced by a point on the circumference of a rolling wheel along a straight line and may be parametrized as $x=t-\sin(t),y=1-\cos(t)$ with t>0.

Just as we use tangent lines to the graph of y=f(x) to determine the rate of change of a function f, we would like to be able to determine how y changes with x when the curve is given by parametric equations. This can be achieved as follows. Let f,g be continuously differentiable functions (their respective first derivatives are continuous). Assume that f′(t)≠0 on some open interval and x=f(t),y=g(t). Then

$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

Polar Coordinates

• Polar to Cartesian: $x=r\cos \theta, y=r\sin\theta$
• Cartesian to Polar: $r^2=x^2+y^2, \tan \theta =\frac{y}{x}$
• x>0 and $\tan\theta=y/x$ implies that $\theta=\arctan(y/x)$
• x=0 and $\tan\theta=y/x$ implies that $\theta=pm \pi/2$
• x<0 and $\tan\theta=y/x$ implies that $\theta=\arctan(y/x)+\pi$

Area in Polar coordinates

If f is a continuous function, then the area bounded by the polar curve $r=f(\theta)$ and the rays $\theta=\alpha, \theta=\beta$  with  $\alpha<\beta$  is given by the following formula

$\frac{1}{2}\int_{\alpha}^{\beta}\left( f\left( \theta\right) \right) ^{2}d\theta$

### Week of  09/25

• Guidelines for Exam 1 (September 28th) This exam will cover Section 11.1, 11.3 and 11.4.
• Study guide Exam 1
• Additional Practices
• Additional video lectures (thanks to the University of Utah)
• Make sure to review all your Blackboard quizzes as well. Additional Practice questions are available on Blackboard. Only attempt the practice when you are done studying for Exam 1. That is first, go over your class notes, study carefully examples given to you in class by your instructor, the additional practice on the class blog, and your Blackboard quizzes. You will not receive a grade for the practice questions and you have unlimited attempts (take advantage of that). In case you do not perform well on the questions, watch the appropriate video lectures located on https://www.math.utah.edu/lectures/math2210.html and make sure you talk to your instructor.

Week of 10/3

• Homework on Section 12.1 and 12.2 posted on Blackboard. It is due on October 7, 2017
• Before you attempt this set of homework, please first go over the additional practice questions and check you answers agains the solutions which are posted on this blog.

### Spring 17 (archived version)

Last week

• Exam 3 (Thu 27 Apr)
• Study Guide posted on the blog
• Final exams
• Final Exam Math 261-001 (Tuesday, May 9 11:00 AM – 1:00 M)
• Final Exam Math 261-002 (Thursday, May 4 11:00 AM – 1:00 PM)
• Project: select 5 questions instead of 10 and solve them.
• Due Wed 04/26/17
• Office Hours Tuesday 04/25/17 from 4:00 PM
• Extra credits for attending Math Chats 04/25/17 and 04/27/17 starting at 3:30 PM

### Exam 3 (study-guide)

13.1 Vector-Valued Functions and 13.2 Calculus of Vector-Valued Functions

• Homework Set 10 (Blog)
• Blackboard Quiz on vector valued functions

14.1 Functions of Two or More Variables

• Homework Set 11 (Blog)
• Quiz on multivariable functions

14.2 Limits and Continuity in Several Variables

• Homework Set 12 (Blog)
• Quiz on Limits and Partial Derivatives

14.3 Partial Derivatives and 14.4 Differentiability and Tangent Planes

14.5 The Gradient and Directional Derivatives

• Homework Set 14 (Blog)
• Quiz on Directional Derivatives and Double Integration

15.1 Integration in Two Variables and 15.2 Double integrals over More General Regions

• Homework Set 16 (Blog)
• Quiz on Directional Derivatives and Double Integration

### Trig Review

Note: Download the files in order to watch the complete videos. Also, it is very important to watch the videos in the order in which they are listed below

### Quizzes

• Blackboard Quiz Basic Trig Review
• Blackboard Quiz Trig Review
• Blackboard Quiz Section 11.1

## Fall 2016 (archived)

Week of Dec 12 (Last week of the semester)

• Wed December 14 is the last day of fall classes.
• Your exams will be returned on Wednesday
• This week, we will complete spherical coordinates
• We might start the section of vector field
• MATH-261-001 (91571) – Final exam is scheduled in the MWF 10: 10-time block, which is Monday, December 19th from 8-10 am in DMF240
• MATH-261-002 (91197) – Final exam is scheduled in the MWF 12: 20-time block, which is Wednesday, December 21st from 11-1 pm DMF259
• You should be working on Homework Set 18
• Find here merged-exams-and-quizzes a single pdf which contains all three exams with their respective study guides as well as all four quizzes (solutions for Quiz 2 and Quiz 4 are missing)

## Handouts

1. introduction
2. syllabus-001
3. syllabus-002
4. section 11.4 (Area in Polar Coordinates)  blank-polar-graphs
5. cartesian-coordinates
6. Section 12.2 lecture-notes-for-students
7. Section 12.3 dot-product
8. Section 12.4 cross-product
9. Section 12.5  section-12-5-students
10. Section 12.7 lecture-notes-12-7-students
11. Section 13.1 section-13-1
12. Section 13.2  section-13-2-students-version
13. Section 14.1 section-14-1-student-version
14. quick-summary-of-concepts-in-vector-calculus
15. Section 14.2 section-14-2
16. Quick-review-for-limits-of-bivariate-functions
17. Section 14.3  partial-derivative
18. Section 14.4 section-14-4
19. Section 14.5 and 14.6 section-14-5-and-14-6   section-14-5-and-14-6-part-ii
20. Section 14.7 section-14-7-part-i
21. Chapter 15

## Trig Review

Note: Download the files in order to watch the complete videos. Also, it is very important to watch the videos in the order in which they are listed below

## Quizzes & Exams

• quiz-1   solutions-quiz-1
• quiz-2 quiz-2-version-2
• quiz-3  solutions-for-quiz-3
• Study guide for exam I review-for-exam-i  exam-i  exam-i-solutions
• Study guide for exam II: review-packet-for-exam-ii  Exam II exam2  exam-2-solutions
• Study guide for exam III review-guide-for-exam-iii

## Homework

• Exercises Manual  (28 pages)
• Addendum
• Supplement Homework on dot and cross products (challenging) dot-product-and-cross-product
• solutions-for-exercises-manual (193 pages)

## Old Announcements

Week 9/26-9/30

We will attempt to get through the following sections

• 3 Dot Product and the Angle between Two Vectors
• 4 The Cross Product
• 5 Planes in Three-Space
• 7 Cylindrical and Spherical Coordinates

Videos

Dot product

Work and dot product

Torque and cross product

Also, do not forget that we have our first exam on Friday 09/30. Please refer to the packet giving to you for additional information.

This exam will cover the following sections

• 1 Parametric Equations
• 3 Polar Coordinates
• 4 Area in Polar Coordinates
• 1 Vectors in the Plane
• 2 Vectors in Three Dimensions
• 3 Dot Product and the Angle between Two Vectors
• 4 The Cross Product

Important Make sure you read your textbook. Your textbook is one of your best resources. It was written to be essentially used by students. The textbook has been carefully written, revised and improved over several years. Make sure you read and work through all relevant examples in your textbook. Moreover, you should also revisit your class notes. Go over all examples done in class, and all exercises assigned. More precisely, you should work out all problems assigned in

• Homework Set 0
• Homework Set 1
• Homework Set 2 (Part I and Part II)
• Homework Set 3
• Homework Set 4
• Homework Set 5
• Homework Set 6
• Homework Set 7

Once you are done studying for the materials above, please go over the additional practice exam questions which are attached to this document. Although, you will be given solutions for all questions, please try to work out the problems without reading the solutions first. Use the solutions to verify that your work is correct. In case you are stuck after trying the same question too many times, you should definitely read the solutions and/or discuss the problems with your professor or a peer.

Week  Sept 12 -Sept 16

• You should have completed homework Set 0 and Homework Set 1 (perhaps still working on part of homework set 1)
• You are being asked to work on Homework Set 2 Part I and Part II
• Quiz 1 is scheduled for Friday, Sept 16 (I will give you more information about the content of the quiz on Wednesday once I have a better sense of the amount of materials which we will cover this week)

Week 10/24-10/28

• Section 14.3
• Section 14.4

Friday, October 21st, 2016

• Quiz 3 is posted on the following link: quiz-3. Question 4 (a) should be changed to f(x,y)=x^2-2y.
• Regarding our third exam which is scheduled for Friday, Oct 28th, a study guide will be available on either Monday or Tuesday.

Week 10/17-10/21

This week, I am hoping to get through Sections 13.2, 14.1 and 14.2

I am assigning the following homework Sets

• Homework Set 11
• Homework Set 12
• Homework Set 13
• Homework Set 14

Week 10/3-10/7

Here is the plan for this week:

• Cover Sections 12.5 Planes in Three-Space, 12.7 Cylindrical and Spherical Coordinates and 13.1 Vector-Valued Functions
• Homework: Homework Set 8 and Homework Set 9 and possibly Homework Set 10
• Additional exercises to test your conceptual understanding of Section 12.5: click here additional-questions-for-section-12-5
• Quiz 2 (the content will be clarified in class) on Friday

Week of Nov 28th-Dec 2

• Additional assignment for this week. It might be very beneficial for you to watch the following videos ahead
•  Integration in polar coordinates
• Changing the order of integration

Projects

• project-1
• project-2
• project-3
• project-4

Friday November 18th

Week of November 1st

• Tentative plan: 14.7 Optimization in Several Variables and 15.1 and 15.2 (double integral)
• Homework sets: You should have completed Homework Set 15. This week, you are being asked to work on Homework Set 16 and homework set 17 Part I

• Project 1
• Project 2
• Project 3
• Project 4
• Project 5
• Project 6

About the Torque

## Trig Review

Note: Download the files in order to watch the complete videos. Also, it is very important to watch the videos in the order in which they are listed below

## Exams and quizzes

• Quiz 1 Solutions
• Quiz 2 Solutions
• Exam 1 Solutions
• Exam 2 Solutions
• Exam 3 Solutions

Illustration: trajectory of a particle on the plane

## Fall 2014 Calculus III

Lectures (videos)

Most of the videos below were recorded by me for this course. I hope that you enjoy them

Announcements

• (Quiz 2) We will have quiz #2 on Friday, October 10th 2014. The quiz will cover Section 11.4 and Section 12.1. Please, review solutions for the homework assigned for the corresponding sections. Stop by my office during my office hours if you have questions.

Syllabus

Lecture notes and Handouts

• Review of Trig
• Mathematica Demonstration of Parametric curves
• Polar Curves
• Notes on unit vectors
• 12.2
• Cylindrical Coordinates and Spherical Coordinates

Quizzes and Exams

• Quiz1
• Exam 1
• Quiz2  Solutions Quiz 2
• Exam 2  Solutions to Exam 2
• Quiz3  Quiz 3 solutions

Homework and Solutions * the links have been removed. Email me if you need access to these documents*

• Homework Set 1 Solutions
• Homework Set 2 and Solutions
• Homework Set 3 and Solutions
• Homework set 4:  Pg 638-640: #1,4,5,7,13,25,27 Solutions to homework for section 11.4
• Homework for Section 12.1 (Vectors in the Plane) Page 665-667 #2, 4, 7, 13, 17, 18, 25, 29, 37, 39, 41, 43, 46, 57 Section 12.1 Solutions
• Homework for Section 12.2 and Solutions to Homework Section 12.2
• Section 12.3
• Section 12.4
• Section 12.5
• Section 12.7
• Solutions to Homework 13.1 and 13.2
• Homework 14.1
• Solutions 14.2
• Solutions 14.3
• Solutions 14.5
• Solutions homework 14.7
• Homework 15.1 and 15.2 Section 15.1 and 15.2

Worksheets and Study-Guides

## Spring 2014 Calculus III

Course Description: This class is a standard multivariable calculus which extends the notion of derivative and integral to higher dimensional spaces. It provides ways of understanding motion on planes and 3-dimensional spaces, motion, speed, acceleration, curvature of smooth curves, heat equation, and forces acting on objects… We will also deal with optimization with and without constraints and computations of volume, surface area of solids and computation of arc length. We will also learn how to use different coordinates systems other than the conventional Cartesian coordinates efficiently. Some examples of different coordinate systems will be polar coordinates, cylindrical coordinates, and spherical coordinates system. Ultimately at the end of this course, your thinking skills, problem-solving skills and ability to visualize higher dimensional objects will be greatly enhanced.

Required Text: Calculus: Early Transcendentals by Rogawski, 2nd edition.

Some videos (by the instructor)

• Finding the volume of a sphere and a cylinder

## Analytic Geometry and Calculus III

This class is a standard multivariable calculus which extends the notion of derivative and integral to higher dimensional spaces. It provides ways of understanding motion on planes and 3-dimensional spaces, motion, speed, acceleration, curvature of smooth curves, heat equation, and forces acting on objects… We will also deal with optimization with and without constraints and computations of volume, surface area of solids and computation of arc length. We will also learn how to use different coordinates systems other than the conventional Cartesian coordinates efficiently. Some examples of different coordinate systems will be polar coordinates, cylindrical coordinates, and spherical coordinates system.  We will learn the use of computer algebra systems such as Mathematica, and how to use Mathematica to solve complicated problems in much easier ways. Ultimately at the end of this course, your thinking skills, problem-solving skills, and ability to visualize higher dimensional objects will be greatly enhanced.

## General Information

Instructor: Vignon S. Oussa
Class time: 6:00 PM -9:50 PM
Class Location: Belleville Main Campus room 1002
Toll Free in Illinois: 1-866-942-SWIC begin_of_the_skype_highlighting            1-866-942-SWIC      end_of_the_skype_highlighting
E-mail: vignon.oussa@swic.edu
Website: http://www.swic.edu

KEY POINTS
A. Vectors and Surfaces
B. Vector-Valued Functions
C. Multivariable Functions
D. Multiple integrals
TEXTBOOK Calculus: Early Transcendentals
Stewart, 5th edition, 2003
Thomson

## Lectures notes & other resources

1. Lecture notes Calc III (please report any typos or mistake to me)
2. Chapter 12 (handwritten lecture notes)
3. Chapter 13 (handwritten lecture notes)
4. Chapter 14 (handwritten lecture notes)
5. Chapter 15 (handwritten lecture notes)
6. Summary of main ideas by Oliver KnillD
7. Online 3D Grapher  (Wolfram Mathematica website)
8. Stewart official web pages
9. Official Mathematica tutorial
10. Maxima free CAS alternative to Mathematica

## Mathematica Resources

You will find here some information which will be useful for your Mathematica project. For first user, download this file. Some of the frequently used commands are:

• D[Sin[x],x] will compute the derivative of $sin(x)$ will respect to x
• Integrate[ x Sin[x], x]  will find the antiderivative of the function $x sin(x)$
• Plot[Sin[x], {x, 0,2Pi}] this command will plot the graph of sine from 0 to 2Pi
• Plot3D[x^2+y^2, {x,-2,2}, {y,-2,2}] to plot the surface in 3 dimension of  $x^2+y^2$
• ParametricPlot[{t, t^3}, {t,-2,2}] will plot the graph of the curve $c(t)=$ in 2 dimension
• ParametricPlot3D[{x, y, x y}, {x,-5,5},{y,-5,5}] will plot the graph of a surface in 3D

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