This page contains past news items previously posted on the homepage (links in items posted before the start of the current term may no longer be valid).
[Nov. 30] IMPORTANT ANNOUNCEMENT:
Today's tutorial on Thursday November 30 has been cancelled. In its place, a final exam review tutorial will be held on:
Friday December 1 at 6:00 P.M. in McConnell 204
We will be reviewing the course and doing a lot of problems in the tutorial; hope to see you all there!
[Nov. 29] Some updates:
This site's address has moved off of McGill's servers to what will eventually become a larger website of mine, so please update your bookmarks.
Tutorials 9 through 12 on tutorials page have been added and/or updated; you may want to download the material in these tutorials again if you previously did, since some of it has changed. In particular, Tutorial 9 now contains extra exercises on the divergence theorem, while the Fourier series material is now in Tutorials 10 through 12. The only thing missing is the solution set for Tutorial 12 (BVPs), which will be posted later.
[Nov. 23] The tutorials page isn't updated and the solutions aren't scanned, but here are the slides for Tutorial 12, which I couldn't show today due to technical problems with the room. Incidentally, I think the recent power outages in the last 3 days may have brought the site down during those times, but the last one was yesterday so that shouldn't be a problem anymore.
Hopefully the slides and the solutions given in tutorial will help with anything you haven't finished in Assignment 6. Some general pointers:
Question 1: Using the heat equation, find the steady state solution (we discussed this in tutorial) and heat flux at the boundary. As far as I can tell, you don't need to fully solve the heat equation to do this question.
Question 2: Apply the method in the slides, using the separation of variables technique to build your own general solution in step 2 (you should have seen in lecture how this is done).
Question 3: Use the even and odd extensions as stated in the slides to find the series that are asked for in this question.
Question 4: The first two are heat equation BVPs, the third is a wave BVP. Again, see the slides for how to proceed.
Expect a larger update after the assignment. Good luck! (Update: I also recommend checking out Assignment 6 of Winter 2006, which is solved, for some additional practice; again, use the method in the slides.)
[Nov. 11] I've posted solutions to Tutorial 9 (previously called Tutorial 10), which contain plenty of extra problems on Fourier series. Be sure you master this material: it's not easy, and it's definitely important. Also check out Assignment 6; you can do Question 3 if you understand Tutorial 9 and define odd and even extensions correctly (I might update the slides and solutions to clarify this later). The remaining questions are on PDEs and BVPs: start them if you feel you're able to, otherwise we'll be covering this topic in the upcoming Tutorial 10.
Tutorial 9 slides have been posted in advance at the tutorials page; it might be good to check them out now, particularly the material it contains on complex numbers if you need some review on that. We'll be using complex numbers extensively in the upcoming part of the course on Fourier series.
The link to the Tutorial 7 slides on the tutorials page has been fixed (it previously pointed to the wrong set of slides).
[Nov. 5] Lots of updates, including some stuff relevant to Assignment 5!
Assignment 2 solutions were updated with figures that were missing in the first version. Some of you have asked if I can put up solutions to the other assignments, but unfortunately this isn't possible since I didn't grade or solve them.
Revamped, easier-to-use Maple code has been posted for Tutorials 1 through 6; check it out at the tutorials page. As I've mentioned before, you can modify this code to verify some of your assignment questions.
The Tutorial 7 slides have been posted; these cover the fundamental theorems of vector calculus, although much of the same is available in the cheat sheet posted earlier.
The Tutorial 5 slides were revamped to simplify some confusion concerning scalar and vector line integrals (this notification was posted earlier also).
Lastly, some hints on the current assignment (more detailed hints were given during my tutorial section last week):
Question 1: Use the divergence theorem. Part A can be done in exactly two short steps; for Part B you'll have to prove a vector identity.
Question 2: Use Stokes' theorem. Part A is straightforward to do with the methods in the tutorial slides; Part B requires a bit of thought, but is not too long or involved once you figure out what to do.
Question 3: Sections 4 and 5.1 of the Wikipedia page on PDEs will tell you everything you need to know to do this question. Even the bonus part isn't too difficult to get if you look the solution up on the Web.
Question 4: Use the multivariable chain rule; if you didn't attend my tutorial section last week, just look it up in a textbook.
[Oct. 25] Further to what's written below, I've also added the Tutorial 6 slides to the tutorials page, which may help with some of Assignment 4. The slides for Tutorial 5 have also been updated to clarify some confusion with scalar versus vector line integrals. As the new set of slides states, all of the forms of line integral are interchangeable: what decides if a line integral is scalar or vector is not the form it's in, but rather if a direction of integration is specified (vector) or not (scalar). The only way this impacts the problem is in how you parametrize the curve of integration, specifically whether the parameter is allowed to go from large to small values or not (it's allowed in vector line integrals, but not in scalar ones).
[Oct. 25] I'd hoped to have an update up sooner (particularly for scans of the last couple of tutorial solutions), but I've been much busier than anticipated lately, and so a more extensive update will likely only come after this assignment. I apologize for this; I'll nevertheless list some tips for Assignment 4 that will hopefully help if you're still stuck on any particular problem.
Question 1, Part A: Showing that the curl of F is zero is straightforward (just work out the cross product in the definition of the curl). Then find a scalar potential function in the manner discussed in Tutorial 6 (that is, use the exact ODEs method to solve F = grad phi).
Question 1, Part B: Straightforward flux integral; set it up and evaluate it using the methods described in Tutorial 4. Note that the surface is already given to you parametrized, so you don't need to do steps 1, 2, and 3 (sketch, choice of coordinate system, and parametrization) in the 6-step method for evaluating integral (described in Tutorial 1).
Question 2: Again, a straightforward flux integral. Just express F first in terms of x, y, and z; recall here that r = x i + y j + z k, where i, j, and k are the three Cartesian unit vectors.
Question 3: The 3D region of integration for this question is not a simple one; try plotting it with Maple if you can't visualize it or doubt your sketch. Basically you have a sphere of radius 3, and are removing the part inside a cylinder of radius sqrt(2). To evaluate the integral, I'd suggest using the divergence theorem to find the flux out of the whole sphere (hint: you won't have to calculate very much at all if you remember what div of a curl is), and then subtracting the flux through the holes in the sphere, which you can evaluate with Stokes' theorem (check the fundamental theorems of calculus cheat sheet) The question looks hard, but actually isn't too bad once you figure out how to approach it.
Question 4: Again, not an easy surface to visualize; I'd recommend looking up the parametrization for a helical ramp and the associated boundary (a helix) in a textbook or on the Web. Once you have the parametrization, just compute the flux through the surface as well as the line integral along the boundary using the standard procedures we covered in tutorial.
As I've said in tutorial, evidently try to do well on your assignments, but remember that your final is worth 90% and will make or break your grade. The assignments are intended to be more of a learning experience and to ensure you don't fall behind, but if you're finding them a bit advanced compared to the material we've covered, don't despair: just do what you reasonably can on them, and stay focused on preparing for that 90% final. This is especially key if the exam retains its current slot at the beginning of the exam schedule, because you won't have too much time to revise in this case.
Good luck on Assignment 4; best wishes for the week.
[Oct. 19] Be sure to start working on Assignment 4. I'll be updating the site in the near future with tutorial material and some advice for the assignment.
[Oct. 10] Finally, some updates! Apologies for not doing this sooner, but just as I'm sure it's been with most of you, work has been a bit (OK, extremely) hectic as of late. In any case, please take note of the following, some of which may help you on Assignment 3 if you're still stuck on anything:
Tutorial 3 and Tutorial 4 material - slides, problems and solutions, as well as Maple code - have now posted on the tutorials page. I've also included a couple of problems not covered in tutorial too, including a solution for Question 4 of Assignment 1 in the Tutorial 3 problems.
Speaking of assignment solutions, I've written up some solutions to Assignment 2 and placed them on the exams and assignments page. The assignments themselves will be graded and returned to you this Thursday (Oct. 12) in class. Incidentally, Question 4 was graded only on participation, and most people got full marks for attempting it, even if the answer was wrong.
Question 1: You need to know how to do line integrals to answer this question. We haven't covered this in tutorial yet unfortunately (the assignments are running ahead of what we can reasonably cover in the tutorials at the moment). To this end, I've posted the notes for Tutorial 5 (the next tutorial) in advance to help you, but you can also check out your class notes or any good textbook for examples (the line integrals are fairly simple ones).
Question 4: Part A is straightforward: simply compute the divergence and apply the method in the Tutorial 1 slides to solve the integral. Part B involves four flux integrals over the four faces of the tetrahedron, but you don't actually need to compute them, if you use the appropriate theorem in the fundamental theorems of calculus cheat sheet along with your result from Part A.
I hope this helps, even if it is a bit late... Good luck on Assignment 3!
[Sept. 21] Tutorial 2 material, i.e. problems and solutions with Maple code, has been posted (the slides are the same as those for Tutorial 1).
[Sept. 17] Some updates relative to the tutorials and your first assignment (sorry for the delay):
Tutorial 1 material - slides, problems and solutions, as well as Maple code - is now posted on the tutorials page. I've also included a couple of problems not covered in tutorial; next week we'll be doing more multiple integration as well.
For Assignment 1, don't forget you can always use Maple (or an equivalent program) to check your answers. Try downloading the code for Tutorial 1 and modifying it as required; it shouldn't take too long.
Finally, some of you have asked about how to approach Question 4 of the assignment. As far as I can tell, solving the question as it's posed requires material we won't be covering until some time from now (namely vector functions and integration over 3D surfaces, which we should get to around Tutorials 5 and 6), so don't worry if you can't understand the hint. You could try looking up the material needed, but it's not simple as it builds on other things we'll be doing soon. My recommendation is to solve the problem by more conventional means, using things you already know - the shape of the region is simple - although this isn't really what the question asks for. At any rate, the problem won't count for much in your final grade, and it's even possible it might not be corrected at all by the marking TA.
[Sept. 13] Hello everyone! I'm the Thursday TA for the Fall 2006 section of MATH 264 Advanced Calculus, and welcome to my tutorials page. On this site you'll find a plethora of useful stuff to assist you during the term, including slides summarizing the important material, worked problems, past exams with solutions, and lots more. The site will be updated throughout the term, so stay tuned, keep working hard, and remember we're always here to help out!
[Apr. 16] This is the last update for this site this term. The solutions for Tutorial 12 are posted, but unfortunately lack any problems on the heat equation. I highly recommend doing Assignment 6 to get some practice with this.
So here we are, the end of another year... or almost, as soon as exams are over. Hopefully you've started reviewing already, but either way, here are my suggestions for what you should be sure to do before the final exam in order to best prepare:
Skim through your class notes, and read through the slides posted in the tutorials section. I highly recommend printing out the two reference sheets on coordinate transforms and vector calculus theorems and memorizing their content. This step serves to make sure you have at your disposal all the material you need to solve problems, and shouldn't take very long since you already know most of it.
Scan through the solutions to some of the tutorial problems, and make sure you can solve them (there are a lot of them though, so I wouldn't recommend spending too much time on this step or getting hung up on some of the finer details in these problems). The important thing is to make sure you have a problem-solving method that works in general: on the exam, you won't have time to think too much or to try things that may or may not work. Again, this step shouldn't take too long, especially if you've been following the tutorials or practicing a lot during the term.
Do as many past final exam problems as you can, starting with the exams that have solutions so you can check your work. For those that don't have solutions, my advice is to use Maple or other software to check your work. The further you go in this step, the higher your chances are of getting a good grade.
Above all, remember that while cramming can work for some courses, it doesn't stand much chance of working in this class. There's simply too much material that requires practice and a solid understanding of the material that came before it. I know this sounds cliche, but it's very true for this class, so be sure to practice a lot, and not just the day before the exam.
In conclusion, I just wanted to say I've greatly enjoyed being your TA this term and working with so many bright students. It was wonderful seeing as well as taking part in the learning process that took place throughout the term. Thanks also for the positive feedback on the tutorials during the semester: having more than 13,000 pageloads on this website over the last four months is pretty amazing (although I should mention 2,000 of those came the day before the midterm...), and I received many appreciative comments and emails.
I wish you best of luck on your finals and all your future endeavors. A bientot! -- Sacha
[Apr. 8] Tutorial 12 solutions are taking a while to post because they require some changes but I'm a bit busy at the moment. They should be up by this weekend; in the meantime, I'd recommend doing Assignment 6 (comes with solutions) and reviewing the rest of the material.
[Mar. 27] Check out the latest material for Tutorial 11, which covers Fourier series. Don't forget to do Assignment 5!
[Mar. 20] The material for Tutorial 10 is posted at the tutorials page. Also, the exams and assignments section now includes the questions and solutions for your midterm.
[Mar. 13] The material for Tutorial 9 is posted at the tutorials page.
[Mar. 6] Posted material for Tutorial 8 (some of it is relevant for the midterm). Good luck on tomorrow's test!
[Mar. 3] Midterm review tutorial post-mortem:
The slides used in the tutorial are essentially those from Tutorials 2 to 6, which are available at the tutorials page.
The questions worked in tutorial, as well as many more final exam and midterm problems are available in the exams and assignments section.
Solutions to problems covered in tutorial will not be posted.
If you read the comment posted earlier this evening about Question 1 of Midterm Fall 2005, Version 1, please disregard what it said: the solution presented in tutorial is correct (to the best of my knowledge). The question asks for the region BOUNDED BELOW by the cone and BOUNDED ABOVE by the sphere (the key word here is "bounded"), so phi does indeed go from 0 to pi/4.
Also, for Question 2 on Version 3 of the same midterm, as some students cleverly pointed out, there's a much easier way to do the question: in this case use the coordinate transformation u = y / x^2 and v = x / y^2. You'd then have to calculate the Jacobian determinant and apply the change of variable formula, as described in the coordinate transformations handout (or Tutorial 2 notes). Always be on the alert for tricks like this specific to the problem!
[Feb. 25] Assignment 3 has been graded; check out the following (also mirrored under the Exams and Assignments section):
Grades list Assignment 3 detailed grades are in the first sheet, while overall grades for Assignments 1 to 3 are in the second sheet of the file (all grades are sorted by the last five digits of your student IDs)
[Feb. 14] Check out the slides and Maple code for Tutorial 6, which covers flux integrals. The solutions will be scanned in soon. You may find Tutorial 5 and 6's code useful for checking your answers to Assignment 3.
[Feb. 6] Tutorial 5 material (slides, Maple code, problems, and solutions) has been posted at the usual page. This tutorial dealt with 3D surface integrals.
[Feb. 6] Tutorial 4's slides and solutions have been updated to clarify an issue a clever student pointed out after tutorial about scalar versus vector integrals. The issue is that if you're integrating f(x, y, z) ds around a curve and the curve has a direction, the integral is actually a vector integral, even though it looks like a scalar one. It turns out though that you can evaluate the integral as you would a scalar one, but with one difference: the parameter t can go from largest to smallest value (which can't happen for scalar line integrals). Details on the slides; apologies for any confusion.
[Jan. 30] Tutorial 4 material (slides, Maple code, problems, and solutions) has been posted at the usual page.
[Jan. 26] Tutorial 3 solutions and Maple code have been posted; apologies for the delay. You may find this particular Maple code set useful for verifying your answers to Assignment 2... (hint, hint).
[Jan. 23] Tutorial 3 slides and problems have been posted at the tutorials page. Maple code and solutions will be posted by tomorrow evening.
[Jan. 18] The Exams and Assignments section is up, featuring exams, midterms, and assignments from previous editions of the course, with solutions where available. Check it out here; happy problem solving!
[Jan. 17] There was a typo in the coordinate transformations sheet posted previously; the formula for phi in the spherical inverse mapping should have read arctan( (sqrt(x^2 + y^2)) / z ); z is not squared. This has been corrected.
[Jan. 16] Posted a summary of what you need to know about coordinate transformations in the Reference Material section; get it here!
Disclaimer: Great effort has been made to ensure the material presented here is factually correct; however, users of this material are fully responsible for verifying all material (theorems, formulas, solutions, etc.) presented on their own. For errors or questions, please send an email to the address indicated above.