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# Calculus III (Math 275), Spring 2012

### Course schedule

The rest of the semester will proceed roughly as follows

 Week #6 Feb. 20-24 Section 12.4 (The cross product) Section 12.5 (Planes in three-space) Section 12.7 (Cylinderical and spherical coordinates) Week #7 Feb 27 - Mar 2 Section 13.1 (Vector-valued functions) Section 13.2 (Calculus of vector-valued functions) Section 13.3 (Arc-length and speed) Week #8 Mar 6 - Mar 9. Midterm #2 Section 13.4 (Curvature) Section 13.5 (Motion in three-space) Week #9 Mar 12 - Mar 16. Section 13.6 (Planetary Motion according to Kepler and Newton) Section 14.1 (Functions of two or more variables) Section 14.2 (Limits and continuity in several variables) Section 14.3 (Partial derivatives) Week #10 Mar 19 - Mar 23 Section 14.4 (Differentiability and tangent planes) Section 14.5 (The gradient and directional derivatives) Section 14.6 (The chain rule) Week #11 Apr 2 - Apr 6 Section 14.7 (Optimization in several variables) Section 15.1 (Integration in two variables) Section 15.2 (Double integrals over more general regions) Week #12 Apr 9 - Apr 13 Section 15.6 (Change of variables) Section 16.1 (Vector fields) Section 16.2 (Line integrals) Week #13 Apr 16 - Apr 20) Section 16.3 (Conservative vector fields) Section 16.4 (Parametrized surfaces and surface integrals) Section 16.5 (Surface integrals of vector fields) Week #14 Apr 23 - Apr 27 Section 17.1 (Green's Theorem) Section 17.2 (Stokes's Theorem) Section 17.3 (Divergence Theorem) Week #15 Apr 30 - May 4 Review Final Wednesday May 9, 10:30AM - 12:30PM The final will be in our regular classroom

### In-class exercises

Below are the in-class exercises and their solutions

 In-class #2 Handed out Tuesday 2/21. Dot-product and cross-product Exercise (in-class #2) Solutions (page 1, page 2) In-class #3 Handed out Wednesday 2/22. Torque Exercise (in-class #3) Solutions (page 1, page 2, page 3, page 4) In-class #4 Handed out Friday 3/2. Arc-length parameterizations Exercise (in-class #4) Solutions (page 1, page 2) In-class #5 Handed out Friday 3/9. Curvature Exercise (in-class #5) In-class #6 Handed out Tuesday 3/13. Vectors and motion, and their connection to curve geometry Exercise (in-class #6) Review sheet on curves Graphs of the helix and the normal and tangent vectors. In-class #7 Handed out Tuesday 4/3. More on level curves, gradients and tangent planes Exercise (in-class #7) Graphs of the functions and the gradient field. Solutions (page 1, page 2) In-class #8 Handed out Monday 4/9. Optimization in two variables, double integrals Exercise (in-class #8) Graphs of the function for problem #1 In-class #9 Handed out Wednesday 4/25. Surface integrals Exercise (in-class #9) Please be prepared in the solution to Problem 2 on Tuesday May 1 Solutions (#2 not worked out completely) (page 1) In-class #10 Handed out Friday 4/27. Surface integrals of vector functions Exercise (in-class #10) Please be prepared to hand this in on Monday, April 30th. Solutions (not worked out) (page 1, page 2) Solutions (complete) (page 1, page 2, page 3, page 4, page 5, page 6) In-class #11 Handed out Wednesday 5/2. Green's Theorem, Stokes' Theorem, and the Divergence Theorem Exercise (in-class #11) Solutions (page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8)

### Homework assignments

Homeworks will count for 10% of your grade and will be done on the textbook website. Homeworks cannot be turned in late.

 Homework #1 Due Monday, January 23, start of class. Please send me an e-mail from an account you use frequently. Put "Math 275" in the subject area. You may leave the message area blank, if you wish. calhoun@math.boisestate.edu Webassign is now setup for this course. Read login instructions here. I will also list problems from the book below. These problems should roughly correspond to those that have been assigned in WebAssign. Section 11.1 : #1, 2, 3, 6, 7, 8, 11, 14, 19, 20, 22, 23, 24, 29, 35, 36, 39, 44 Homework #2 Due Friday, January 27, start of class. You will do your problems on WebAssign. But if you'd like extra practice, you can also do the following problems from the book. I will not collect these extra problems, but certainly ask about them if you have questions. Section 11.1 : #26, 42, 49, 50, 53, 57, 58, 62, 65, 71, 80. Section 11.2 : #1, 6, 7, 9, 10, 11, 15, 18, 20. Section 11.3 : #1, 5, 6, 11, 14. Homework #3 Due Friday, February 3. Section 11.3 : #8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 36, 45. Section 11.4 : #11, 12, 13, 15, 21, 23, 24, 25, 26, 29 Section 11.5 : #1, 13, 15, 16, 17, 39, 40, 42, 43. Homework #4 Due Monday, February 13. Section 12.1 : #3, 6, 7, 9, 23, 24, 25, 28, 31, 32, 33, 35, 37, 39, 41, 43, 45, 48, 50, 54 Section 12.2 : #1, 6, 21, 25, 27, 29, 30, 31, 32, 34, 38, 40, 50, 56 Homework #5 Due Tuesday, February 21. Section 12.3 : #2, 6, 7, 8, 9, 10, 13, 14, 18, 19, 20, 25, 26, 28, 29, 33, 34, 36, 40 Section 12.4 : #1, 5, 9, 13, 14, 17, 23, 24, 30, 35, 43, 59, 60, 61 Homework #6 Due Monday, February 27. Section 12.5 : #1, 2, 5, 8, 11, 13, 15, 17, 21, 27, 29, 34, 37, 49, 54, 61, 64 Section 12.7 : #1, 4, 5, 6, 25, 26, 31, 32, 33, 36, 39, 61, 65, 67, 69, 70, 71 Homework #7 Due Monday, March 5. Section 13.1 : #1, 3, 4, 8, 9, 10, 11, 15, 19, 21, 22, 23, 25, 26, 27, 33, 34, 36 Section 13.2 : #7, 11, 14, 16, 17, 18, 19, 20, 23, 29, 30, 55, 56 Section 13.3 : #1, 2, 3, 7, 8, 10, 12, 18, 22, 27, 29 Homework #8 Due Monday, March 12. Section 13.4 : #1, 5, 7, 9, 12, 15, 19, 30, 36, 37, 44, 46, 59, 68. Section 13.5 : #3, 5, 12, 15, 17, 19, 25, 26, 28, 30, 33, 34, 43, 44, 45, 47. Homework #9 Due Thursday, March 22. Section 14.1 : #2, 13, 18, 19, 20, 21, 23, 24, 29, 30, 31, 32, 34, 35, 36, 40, 41, 43, 47. Section 14.2 : #1, 4, 12, 21, 30, 34 Section 14.3 : #3, 5, 10, 18, 22, 25, 26, 27, 29, 34, 36, 37, 47, 48, 57, 58, 59, 61, 62, 68 Homework #10 Due Tuesday, April 3. Section 14.4 : #1, 7, 11, 15, 18. Section 14.5 : #30, 32, 33, 36, 37, 38, 44, 58 Section 14.6 : #1, 3, 5, 10, 14. Homework #11 Due Monday, April 16. Section 14.6 : #2, 4, 6, 9, 13, 26, 27, 37 Section 14.7 : #4, 5, 6, 8, 10, 11, 15 Section 15.1 : #3, 8, 9, 19, 21, 24, 36 Section 15.2 : #3, 4, 6, 8, 20 Section 15.3 : #3, 4, 6, 8, 20 Section 15.4 : #1, 2, 4, 6, 8, 13 Section 15.6 : #24, 25, 30, 35 Homework #12 Due Monday, April 23. Section 16.1 : #14, 19, 27, 29 Section 16.2 : #2, 15, 5,6,7,8 Homework #13 Due Wednesday, April 25. Section 16.3 : #2, 6, 11 12, 23, Homework #14 Due Monday, April 30 Section 16.4 : #1, 3, 9, 19, 22 Section 16.5 : #1, 6, 10, 13 Go to WebAssign WebAssign

### Quizzes

Quizzes and shorter in-class exercises will be worth 10% of your grade.

 Quiz #1 Friday, January 27. Solutions (page 1). Quiz #2 Friday, February 3. Solutions (.pdf). Quiz #3 Friday, February 17. Solutions (page 1, page 2). Quiz #4 Monday February 27. The quiz will cover material from Chapter 12, including the spherical and cylindrical coordinates. Solutions (page 1, page 2) Quiz #5 Friday March 23. Solutions (page 1, page 2) Quiz #6 Monday, April 16th. Material will cover Sections 14.5, 14.6, 14.7, and Sections 15.1, 15.2, 15.3, 15.4 and 15.6 Solutions (page 1, page 2)

### Midterms

Midterms will be held in our regular classroom, during the regular class meeting time. Please show up on time so you have the full class period to take the exam. Each midterm is worth 20% of your grade.

 Midterm #1 Wednesday, February 8. Material will be from Chapter 11 and Sections 1 and 2 of Chaper 12. Solutions (page 1, page 2, page 3, page 4, page 5, page 6) Midterm #2 Wednesday, March 7. Material will cover sections 12.3 through 13.3 (excluding 12.6). Review sheet Solutions (page 1, page 2, page 3, page 4, page 5) Midterm #3 Wednesday, April 18. For this exam, you will be allowed one page of notes. Front and back side of a regular 8.5 x 11 sheet of paper. Material will cover the following sections Section 13.4 (Curvature) and 13.5 (Motion in three space) Section 14.1 (Differentiation in several variables), 14.3 (Partial derivatives), 14.4 (Tangent planes), 14.5 (The Gradient and Directional Derivative), 14.6 (The Chain Rule), 14.7 (Optimization in Several variables) Section 15.1 (Integration in Two variables), Section 15.2 (Integration over more general regions), 15.4 (Polar coordinates), and Section 15.6 (Change of variables). Review of topics to be covered on the exam. Change of Variables problems for linear maps.

### Final

The course final will be on May 9th, 10:30AM - 12:30PM in our regular classroom.

 Final Wednesday, May 9, 10:30AM-12:30PM. The final is in our regularly scheduled classroom To review for the final, please go to a set of problems that have been posted on WebAssign I also have a set of review problems that you can practice here. You will be allowed one sheet of notes - both sides of an 8.5 x 11 sheet of paper. The final will cover material from the following sections Section 12.1 (Vectors in the plane), 12.2 (Vectors in three dimensions), 12.3 (Dot product and and angle between two vectors), 12.4 (The cross product), 12.5 (Planes in three-space) Section 13.1 (Vector-valued functions), 13.2 (Calculus of vector valued functions), 13.3 (Arc-length and speed), 13.4 (curvature), 13.5 (Motion in three-space). Section 14.1 (Functions of two or more variables), Section 14.3 (partial derivatives), Section 14.4 (Differentiability and tangent planes), 14.5 (Gradient and directional derivative), 14.6 (The chain rule), 14.7 (optimization in several variables) Section 15.1 (integration in two variables), 15.2 (double integrals over more general regions), 15.6 (Change of variables) 16.1 (Vector fields), 16.2 (line integrals), 16.3 (conservative vector fields), 16.4 (Parameterized surfaces and surface integrals), 16.5 (surface integrals of vector fields) 17.1 (Green's Theorem), 17.2 (Stokes' Theorem), 17.3 (Diverence Theorem). You should be comfortable with the following ideas and be able to do the following Be able to parameterize simple curves (lines, circles, ellipses, hyperbolas) in two and three dimensions Know and be able to apply the properties of the dot product and the cross product. You should also understand the difference between computing the dot or cross product, and applying the properties in different situations. Be able to compute the angle between two vectors Understand how the dot product is connected to the magnitude of a vector Understand how the dot product is connected to the angle between two vectors. Understand how the cross product is connected to the angle between two vectors. What is the angle between the cross product vector and the two vectors themselves? Understand what it means to project one vector onto a second vector Be able to find the equation of a plane through three points Be able to identify a normal to a plane, find the intersection of a plane and a line, find the equation of line formed by the intersection of two planes (for simple cases). Find a tangent plane approximation to a function at a given point Understand how unit tangent and unit normal vectors to a given curve in two or three dimensions can be computed from a parametric representation of the curve. Know how to compute arc-length (i.e. distance traveled) along a two or three dimensional curve. Understand how this is related to the speed of a particle whose path is described by a particular set of parametric equations Be able to compute an arc-length parameterization from a given set of parametric equations. What is the key property of an an arc-length paramterization? Be able to compute curvature for two and three dimensional curves Be able to compute the velocity and acceleration vectors from a given position vector in two and three dimensions Decompose the velocity and acceleration vectors into normal and tangential components Be able to compute level curves of functions in two variables Understand the relationship between the gradient vector, level curves, and the directional derivative. What is the derivative of a function along level curves? In what direction is the function increasing or decreasing the fastest? Be able to compute the derivative of a scalar function along a given parameterized curve Be able to apply the chain rule in several variables. Find critical points, and apply the Second Derivative test to find minima, maxima and saddle points of functions of two variables. Be able to integrate functions of two variables over rectangular regions Be able to integrate functions of two variables over more general regions (circles, regions described by functions f(x), g(x), etc). Apply the Change-of-Variables formula to general regions to simplify the integration. This includes the change of variables to convert to polar coordinates Know what a conservative vector field is. How does it simplify the task of computing the circulation around a closed loop? Be able to set up and evaluate the integral of a scalar valued function over a curve in two or three dimensions Be able to setup and evaluate the integral of the projection of a vector field onto unit tangent and unit normal vectors to a curve on two and three dimensions Know what circulation means and how to compute it, given a conservative or non-conservative vector field Given the parameterization of a surface, compute the integral of a scalar valued function in (x,y,z) over a surface Evaluate the integral of the projection of a given vector field onto the normals of a given surface Know what if means to verify Green's Theorem, Stokes' Theorem and the Divergence Theorem (in 2d). Be able to compute the curl and divergence of a vector field. Know which leads to another vector, and which leads to a scalar quantity. Be able to state in words what Green's, Stokes' and the Divergence Theorem are saying.

## Other useful information

### WebAssign is now setup for this course (posted 1/19/2012)

• Your username for this site will be the first letter of your first name, the first letter of your middle name (if available) and your last name. Example : The username for "John T. Smith" is "jtsmith". Please send me e-mail if you have trouble logging in.
• You will now be asked to either pay for access now, or wait until to pay once the grace period is over.
• Once you have logged in, you can change your password by going to "My Options".
• The details above are available as a PDF file here.

Once you are logged in, you will have access to the homework assignment "Homework #1". Good luck! And please be sure to e-mail me if you have questions

### Online-homework system (posted 1/16/2012)

Homeworks will be done on-line via WebAssign. As soon as I get the course set up, you will be able to then go on line (with details to come) and purchase access to the course and all assignments.

### Send me an e-mail! (posted 1/16/2012)

Please send me an e-mail at calhoun@math.boisestate.edu so that I can compile an e-mail list for the class. At the very least, include a subject header that says "Math 275". You may leave the message area blank.

### My office hours (posted 1/16/2012)

My office hours for this semester will be Wednesday 11:30-12:30, or by special appointment.

### Course textbook (posted 1/16/2012)

Our course text books is "Calculus Early Transcendentals (Multivariable)", second edition, by Jon Rogawski. This textbook is being used on a trial basis, and so you will be loaned a copy of the textbook to use for the semester at no cost. You will have to return your copy at the end of the semester. Other calculus courses are using the free online text by Whitman. This text is also available for you to use as a supplement to the Rogawski text.

You may also be interested in the online version of our text. When you purchase access to the WebAssign on-line homework system, you will also have the option of paying for access to the eBook version of the textbook.