Review Information for the Final Exam Spring '02

The final exam will be held on Thursday, May 16th at 2:15pm in Science Center Hall C

There will be several coursewide reviews held by the TFs as well as reviews hosted by CAs, scheduled as follows:

Wed. May 8th  12-2pm  Science Center Hall A        (for all sections)
        covering material from the first part of the semester up through midterm 1
Friday May 10th  2-4pm  Science Center Hall E        (for all sections)
        covering the material that tested on midterm two
Monday May 13th 3-5pm  Science Center Hall A (for non-biochem sections)
        covering the vector calculus material (chapter 13).
        (Melissa's biochem section will have its own review in B11A from 3 to 5pm on the same Monday)

There are five CA problem session reviews scheduled, as follows:
     Tuesday, May 7th,  5-6pm in SC 209 with Gloria Hou
     Wednesday May 8th, 7-8 pm in SC 310 with David Shim
     Sunday, May 12th, 6-7pm in SC 309 with Alexey Gorshkov
     Monday, May 13th, 3-4pm in SC 309 with Roger Hong
     Tuesday, May 14th, 5-6:30pm in SC 309 with Nathan Moore

In addition, there are numerous office hours offered by the TFs as follows: (these are simply the ones announced so far, feel free to get in touch with your TF directly to set up other times)

Spiro (Science Center 421g)
Tue. 5/7 2-5 pm
Wed. 5/8 2-5pm

Melissa (Science Center 536)
Tue.5/7   4-5pm
Thu.5/9   4-5pm
Tue.5/14  3-5pm
Wed.5/15  3-5pm

Andy (Science Center 435)
Tue 5/7 2:30-4
Thur 5/9 2:30-4
Tue 5/14 1-3pm
Wed 5/15 1-3pm

Christophe - meetings by appointment

For the final exam, the problems will cover the whole semester's worth of material.  Naturally there will be an emphasis on the most recent topics as they haven't been tested as of yet.  The breakdown of questions is likely to be about 40% on topics since the second midterm, 30% on material covered on the second midterm, 20% from the first midterm's topics, and about 10% (probably just one shorter question) on the last week's topic of PDEs.  Remember that the final is a three hour final, so there will be more questions on the final (probably around 10 questions or so), than there were on the midterms.

Below please find a list to help you work out what will be covered on this midterm – since the final includes topics from the first two midterms, then instead of repeating the exact same lists from the two previous midterm review sites, we've simply included the links to the first two midterm review sites for you to take a look at again.  Make sure you study your midterm exams and check the solutions (Midterm 1 Spring 2002, Midterm 2 Spring 2002), if  you haven't already done so!

There is a lot of material to study for the final.   It's easy to get overwhelmed if you try to do it all at once, so be sure to be systematic in your approach, and work the semester's material topic by topic.  Pace yourself, and study a bit more each day - don't leave it off until the last minute - it's very unlikely to work out well for you if you do that!  Again, to get ready for a math test be sure to spend as much time as possible practicing doing math – go back to your homework sets and try redoing problems pulled out at random.

On the other hand, you don't want to end up just being able to do the specific practice problems you do, you also want to make sure that you really understand what you're doing, so that if you see variations of problems, that you'll be able to handle solving them as well.  For instance, if you've just figured out how to compute triple integrals, but you don't really understand what they mean, then if you see a problem that says "a unit box located in the first octant with one corner at the origin has a variable density of f(x,y,z).  Find its mass"  then you might not figure out that it's simply asking you to compute the triple integral of  f(x,y,z) over the region of the unit box.

Topics for the final exam include everything from midterm 1 (so please see Midterm 1 Review again), as well as everything from midterm 2 (please see Midterm 2 Review).  In terms of the latest material, covered since the last midterm, then please check the following list

Note, this is just for the Regular and Physics sections - those in the Biochem section should check with Melissa or go to her review session on Monday, May 13th, from 3 to 5 in Science Center B11a
Here are the answers to the extra problems Melissa suggested from Rosner Extra Problems Solutions

Vector Calculus, Stewart Chapter 13:

• Vector Fields
• Graphing vector fields
• Definition of a conservative vector field (equals the gradient of a function)
• Line Integrals
• Definitions and meanings - there are two types: line integral of a function along a curve ("line integral with respect to arc length"),  and a line integral over a vector field (a "work" integral)
• Know how to compute each type (equation 3 on page 925 and equation 13 on page 932)
• Note the variety of notations for line integrals (such as the one at the bottom of page 933)
• Surface Integrals
• Definitions and meanings - again, there are two types:  integrals of a function over a surface (which gives the mass of a surface if the function gives the density at different points), and so-called "flux" or flow integrals, that give the flow of a vector field through a surface.
• Know how to compute each type (equation 2 on page 960, and equation 9 on page 967 respectively)
• Note that to deal with the idea of flow in a particular direction, then the surface needs to have a direction associated with going through it (a particular choice for the outward pointing normal - there are two choices).  For closed surfaces (for instance a sphere), "positive orientation" means an outward pointing normal.
• Also, to deal with surface integrals you need to be able to parametrize the surface in question, so review how to deal with basic surfaces, such as planes, surfaces of graphs, and surfaces of rotation (take a look at this on page 740 again).  Also, for surfaces given by the graph of a function, z = f(x,y), know the shortcut for finding | r_u x r_v | given on top of page 962, so you don't have to compute it each time.
• Curl and Divergence
• Know how to compute each, and have an understanding of what each measures (take a look at the top of page 955 for curl, and the bottom of page 982/983 for divergence)
• Know that the curl of a conservative vector field is the zero vector
• Integral Theorems!
• There are quite a few, so spend some time writing each one done to try to remember what each is used for
• Fundamental Theorem of Line Integrals
• Conservative vector fields are great!  Integrals around closed paths equal zero in a conservative vector field, and integrals from one point to another are path independent, and can be calculated by finding the difference in the values of the potential function at the endpoints.
• Green's Theorem - for two-dimensional line integrals around closed paths (paths that make complete circuits)
• When a vector field isn't conservative then you can convert a line integral into a double integral (look at the equation on page 945 - remember that P and Q are the two components of the vector field that the line integral is being done over, i.e. F = <P, Q> or F = Pi + Qj
• Note, you need to go around the path in the counterclockwise (positive orientation) direction
• Stoke's Theorem - for three-dimensional closed path line integrals
• Generalization of Green's Theorem - note that the integrand in Green's Theorem looks look one part of the definition of curl
• Now a closed path line integral in 3-space can be replaced by a surface integral involving the curl of a vector field (see equation at bottom of page 971) - here it's not always obvious which one is easier to compute, they can both be pretty annoying!  But in any case, you should know how to convert from one to another.  Be sure to take a look at the homework from this section again to remind yourself how to deal with these conversions, and try to figure out which integrals are easier in given situations.
• Divergence Theorem
• A nice theorem to use when you can, it says that the flux (surface) integral of a vector field over a closed surface (with outward pointing normal, so this measures the amount of material flowing out of the region inside the surface) is equal to the divergence of the vector field over the interior of the region (which measures the amount of material being created (sources) or destroyed (sinks)).
• This means that the surface integral can be replaced with an often times easier to deal with triple integral (take a look at the equation on page 978)
• Note there are several topics in the textbook that you don't need to know about:
• No conservation of Energy computations (page 942, section 13.3)
• You don't need to know how to prove each of the integral theorems, just know how to use them
• Vector forms of Green's Theorem (page 957 in section 13.5)
• Skip the Oriented Surfaces section on pages 964-965 in section 13.6 - be assured that the surfaces you have to deal with on the final will be orientable (no Mobius strips!)
• Finally, be sure to go over the PDE handout that covered the material learned in the last week of class (Physics section - check with Spiro about what you'll need to know from your last week of classes).  Everyone else, to prepare for the PDE question on the final (there will likely be one relatively short question on the final on the PDE material), make sure you go through the suggested problems at the back of the handout.
• PDE Handout Suggested Problem Solutions (note in the solutions, the answer to #2(d) should have c = 1, not 100, and in #5(d) B should be -(1/30), not -0.2)
• Know what a PDE is,
• Know some examples of PDEs,
• Know how to check if a function is a solution to a PDE (as in problems 1 and 2 in the suggested problems)
• Know what boundary conditions are and how to check to see if a solution satisfies them,
• Know how to create a specific solution to a given PDE with specific boundary and initial conditions, if you are given a general solution to the PDE (as in problem 5 in the suggested problems)
• Know how to use the separation of variables technique to solve certain basic types of PDEs (as in question 6)

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Old Exams for practise:

• Here are a number of old exams to use for practise.  Remember that to use these effectively, you should really think hard about how you're solving the problems - you're not going to see these exact problems on our final, so you should be concentrating on the process of solving these problems, so you can be better prepared when you see different problems on the same topics.  We will be posting more solutions to these by around Monday, the 13th, but be sure to answer the problems on your own before you check the solutions!
• Final Fall 99 Exam  (Final Fall 99 Solutions)
• Final Fall 00 Practice Exam  (Final Fall 00 Practice Solutions)
• Final Fall 00 Exam (Final Fall 00 Solutions)
• Final Spring 00 Exam (Final Spring 00 Solutions)
• Final Spring 01 Exam (Final Spring 01 Solutions)

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Review Problems for Chapter 13 from our textbook

• The review problems at the end of chapter 13 are pretty good problems for you to use to practice up on all of the different types of integrals we learned about in the last section of the course (unless, of course, you were in Melissa's Biochem section - you will need to work with the Rosner textbook to review for that section).  At this point, you should be able to work on any of the chapter 13 review problems.
• Chapter 13 Review Problem Solutions