Review Information for the Final Exam Fall '03

The final exam will be held on Tuesday, January 27th at 9:15am.  This is a three hour, cumulative exam (i.e. it will have problems covering the whole semester's worth of material).  The class is split into three locations for the final:

 2 Div Ave, #18 (last names starting with A - K)

 William James 1 (last names L - R)

 Harvard 201 (last names S - Z)

Please make sure you know where these places are ahead of time, before you need to go to the exam on Tuesday morning!

As was the case for the earlier midterms, there are no calculators or notes allowed during the final exam.

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

Note if you missed these reviews from last week, then you can check them out on video in Cabot Science Library, or online at http://www.courses.fas.harvard.edu/~math21a/tool/video/

Monday. Jan 12th,  4-5:30pm  Science Center Hall D (for all sections)
        covering material from the first part of the semester up through midterm 1
Tuesday, Jan. 13th,  4-5:30pm  Science Center Hall D  (for all sections)
        covering the material that tested on midterm two
Wednesday. Jan 14th,  4-5:30pm Science Center Hall D (for all sections)
       covering Chapter 12
Thursday,  Jan 15th,  4-5:30pm  Science Center Hall A (note room) for non-biochem sections
        covering the vector calculus material (Chapter 13).
        (Jon and Vivek's biochem sections will have thier own review on Thursday, the 15th, 4-5:30pm, in Science Center E)

There will also be CA problem session reviews to be scheduled during the Reading Period as well

In addition, TFs will be holding office hours throughout Reading Period up until the time of our exam (feel free to get in touch with your TF directly to set up other times).  The following office hours have been scheduled (your TF might have told you of more office hours as well):

Tom Coates SC 334 - MWF 12-1pm
Abhinav Kumar  SC 431d  - MWF (19th/21st/23rd/26th) 4 - 5pm
Andy Engelward  SC 435 -  Monday the 26th from 12:30 to 1:30
Vivek Mohta in Loker Commons - Mon, Tues, Wed 6:30-8:30pm
Namhoon Kim SC 324d  - MWF 2 - 3pm
Teruyoshi Yoshida SC 425d  Wed/Thur/Fri 2-3pm
Sabin Cautis SC 428f - Wed (14th) 8-9pm and Wed(21st) 7-9pm
Jonathan Kaplan SC428b - Tue (20th) through Fri (23rd) 2-4pm

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 50% on topics since the second midterm, 30% on material covered on the second midterm, and 20% from the first midterm's topics.  Remember that the final is a three hour final, so there will be more questions on the final than there were on the midterms (probably around 10 questions or so).

Below please find a list to help you work out what will be covered on the final – 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 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 through 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 covering topics from chapters 12 and 13:

Chapter 12

Double Integrals

• Understanding of basic concept behind definitions – analogy to single variable calculus of Riemann Sums
• Computation of double integrals over various regions through use of iterated integrals (when f is continuous over the region, i.e. Fubini’s Theorem)
• Ability to find limits of integration for various regions (should be able to switch order of integration through recalculating limits - see example 5 on page 859)
• Understanding of double integral as giving volume over a region, under a function of two variables (when the function is positive, in any case. When a function is negative, the double integral will produce a negative result, just as in the single variable case)
• Calculating double integrals with polar coordinates
• Sub in x = rcos(theta), y =rsin(theta) into whatever function f(x,y) is in the integrand
• Replace dA (which was equal to dxdy in rectangular/cartesian coordinates) with rdrd(theta)
• Redescribe the region in terms of the polar coordinates r and theta

Skip section 12.5 (center of mass, moments of inertia, probability and expected values – again, useful to know about, but won’t be tested on our midterm)

Triple Integrals

• Again, a basic understanding of the concept behind the Riemann Sum definition
• Fubini’s Theorem for computation of triple integrals over various volume regions (so for triple integrals now the region of integration is a volume in space, not an area in the xy-plane)
• Ability to find limits of integration for various regions (again, should be able to switch order of integration through recalculating limits - see homework problem 30 on page 892 for a good example)
• Calculating triple integrals with cylindrical coordinates (basically same maneuvers as with polar coordinate conversion, with the additional z coordinate):
• Sub in x = rcos(theta), y =rsin(theta), z=z (!) into whatever function f(x,y,z) is in the integrand
• Replace dV (which was equal to dxdydz in rectangular/cartesian coordinates) with rdrd(theta)dz
• Redescribe the region in terms of the cylindrical coordinates r, (theta), and z.
• Calculating triple integrals with spherical coordinates
• Sub in x = rcos(theta)sin(phi), y =rsin(theta)sin(phi), z=rcos(phi), into whatever function f(x,y,z) is in the integrand
• Replace dV (which was equal to dxdydz in rectangular/cartesian coordinates) with r²sin(phi)drd(theta)d(phi)
• Redescribe the region in terms of the cylindrical coordinates r, theta and phi.

Surface Area computations

• Be able to use the formula for surface area for a parametrized surface r(u,v):
• A(S) =  | r_u x r_v | dudv
• Note that there will be times when you need to come up with the parametrization (such as in problem 22 in section 12.6, and other times when you will be given the parametrization, as in problem 8 in section 12.6 – be prepared to be able to deal with either one!)

Note there are several topics in chapter 12 that will not be covered on this midterm:

• Skip the integration applications in section 12.5
• Skip the several applications of triple integrals in section 12.8 (bottom of page 888 through page 890)
• The final will not cover section 12.9, which we skipped during the fall.

Chapter 13:
Note, this is just for the Regular and Physics sections - those in the Biochem section should check with Jon and Vivek about material for the last three weeks for the semester.

• 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 chapter 13 that you don't need to know about:
• No conservation of energy computations (page 942, section 13.3)
• You don't need to know the proofs of each of the integral theorems, just 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!)
• Also, there will be no questions on PDE's on the final (a separate section not covered in the textbook).

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Old Exams for practise:

• Here are a number of old exams to use for practice.  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 same problems on our final, so you should be concentrating on the process of solving these problems to be better prepared when you see different problems on the same topics.  Note that there are separate problems for the Biochem sections to answer, and these will likely be quite different this semester, so check with Jon and Vivek before assuming you're supposed to know how to answer these practice problems.  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)
• Final Spring 02 Exam (Final Spring 02 Solutions) and Brad's Comments

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Review Problems for Chapters 12 and 13 from our textbook

• The review problems at the end of these chapters 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 a Biochem section for chapter 13).  At this point, you should be able to work through #3 - 34, 38(a), 39, 41, 42, 47 and 48 in the Chapter 12 review, and any of the chapter 13 review problems.
• Chapter 12 Review Problem Solutions and Concept Check
• Chapter 13 Review Problem Solutions and Concept Check