# Math 226: Multivariable Calculus

## Spring 2004

 Instructor: Martha Wallace Find me in OMH 103 Phone me at x3408 Email me Text: Ostebee and Zorn, Multivariable Calculus, (Vol. 3), 2nd edition

What is Multivariable Calculus?

Multivariable Calculus is an extension of the ideas that you studied in Calculus I and II.  Those courses addressed single variable functions that could be graphed in two dimensions, such as f(x)=sin(x).  In Multi you will investigate functions of two variables whose graphs need three dimensions, such as f(x,y)=cos(x+y).  Earlier you learned about and applied derivatives and integrals of functions of a single variable.  In this course we will solve much more interesting problems by finding derivatives and integrals of  functions of two (and sometimes more) variables.  We will compare meanings and techniques between single and multivariable derivatives and integrals, and will use vectors and polar coordinates.  You will see why the concept of multiple representations is a great problem solving tool. We will use Maple 8 software to help us visualize functions in three dimensions. Maple is available only on the computers in SC 175 and OMH 108.

In this class, we will study:
1.  Curves and Vectors (Chapter 12): This chapter extends function descriptions and graphing ideas from the two-dimensional space you studied in first-year calculus to three- (and more) dimensional space.  You will use concepts that you studied in linear algebra, such as space vectors and parametic equations.  You will model motion using derivatives and integrals of vector-valued functions.  You will use vector dot products to find projections of one vector onto another and cross products of three dimensional vectors to find areas and describe planes.
2. Derivatives (Chapter 13): This chapter extends the notion of derivative from two-dimensional space to multidimensioned space.  You will investigate the graphs of multivariate functions through contour maps and optimize such functions by using partial derivatives.  As in first-year calculus, you will look for linear approximations to curves (recall "locally linear") and tangent lines, but you the thinking will be a bit more complex.  You will find the gradient of a function at a point and look at directional derivatives.  You will be able to extend you ideas of higher level derivatives and Taylor polynomials, and will find new beauty in the chain rule.
3. Integrals (Chapter 14): This chapter extends the study of integrals to multivariate functions.  You will approximate integrals by finding the limit of approximating sums just as you did in elementary calculus, only with more complicated regions, resulting in multiple integrals.  You will change rectangular coordinates to polar, cylindrical or spherical coordinates to make your integration tasks easier.
4. Applications and Extensions (Chapter 15):  You will sample these topics through both class and student group work.
5. Vector Calculus (Chapter 16): This chapter develops tools used in physics and other fields that use vector-valued functions.  You will investigate vector fields and line and surface integrals.  You will revisit the Fundamental Theorem of Calculus and pull the concepts of the course together through Green's Theorem.

Class Policies:

Homework Policy:

With few exceptions, you will have two assignments due each day:

A writing assignment based on the material discussed in the previous class as well as often some preview problems from the next sections and possibly some review problems from previous sections. This assignment should be done in draft form by the next class day to allow for a small amount of explication in class. The final form of each assignment is due on the second class day after it is assigned. You are encouraged to work with other class members to do your homework assignments, and may if you wish, submit one paper for two people. (If you do this, be sure to put the names of both contributors on the paper and take turns writing the final draft so that you both get your writing critiqued.) The writing assignments will be corrected and the grades will count toward your final grade in the course.

No late homework will be accepted, but 3 writing homework scores and 3 card quiz scores will be dropped.

Computer Work:
During the semester, we will have a few computer labs and computer components of many other assignments. You will use the computer algebra system Maple 8 for these assignements. This program is available on the computers in SC 175 and OMH 108.   Some of the tests may have a take-home portion on which you will be expected to use Maple.  Maple 8 was new at St. Olaf last semester -- if you have not used it, you may want to read the  Introduction to Maple for Math 226.

 How does your work contribute to your  final grade? What grade will you get in this class? Components: Points Possible: || || Total Points Earned as % of  Possible Minimum Grade You Will Earn Homework, Labs and Quizzes 100-150 points || || 90% A- 2 Tests 200 points || || 80% B- Final 150 points || || 65% C- Total Possible 450-500 points || ||

Hints for Success:

Reading the material carefully before it is covered in class is a big step toward success in any math course.  Successful students typically outline or otherwise summarize the material briefly in their notebook and highlight questions to bring to class.  A great way to become familiar with concepts and techniques is to work each of the examples.  (This means work on paper -- don't just read and nod.)

Make sure that you begin the assigned homework as soon as possible after it is assigned and bring a nearly complete homework paper to the following class so that you can get the most out of any homework discussion in class.  Be sure to make connections in your mind between multivariable concepts and the single variable calculus that you know.

A sticking point for many students in Multivariable is the difficulty in visualizing three dimensional graphs, so be sure to use Maple to help you picture such functions.

Disability Policy:

If you have a documented disability that will impact your work in this class, please contact me to discuss your needs. Additionally, you will need to register with Student Disability Services located at the Academic Support Center in Room 1 of the Old Main Annex. All such discussions will be confidential.

Disclaimer