Fall 2010 UBC

Archive for September, 2010

Concerning your Comments

Thank you for all of your comments.  I appreciate the compliments and encouragements, and also the concerns and suggestions.  I will mostly reply to concerns and suggestions here.  I always welcome more feedback.

“It would be good to have more/harder examples in class.”

Some students asked for fewer examples, but they were outnumbered by those who asked for more or more difficult examples.  I would like to provide more difficult examples, and will try to do so.  The order of priority and presentation is usually: 1) concept; 2) simple example; 3) more complex example.  I don’t think this order can be rearranged, so the complex examples often get dropped in consideration of time.  Consider: this is not necessarily a bad thing, as students will learn more from doing more difficult examples themselves as homework.  Nevertheless, the more complex examples will increase as the course progresses, and I’ll try to choose very interesting and enlightening ones.  I have noticed a few times now where I could have improved an example chosen for class, and I will keep this suggestion in mind in future.

Finally, I will just note that it is good that homework problems challenge you beyond what is covered in lecture, to synthesize the basic concepts into more nuanced understanding.  Each step in the learning process should take you further, instead of just repeat.

“Less emphasis on proof.”

There has been some emphasis at the beginning of class on proof.  That’s because some proofs are required (yes, there always will be some, as these are the best, most challenging way to assess understanding of concepts), and students always find them difficult.  I intend to spend more time on proofs in the beginning, because I want to jump start you all into careful and precise thinking about mathematical concepts.  Proof is not separate from problem solving: in fact, they both rely on the same skills.  Real (instead of cooked up) problems that can be solved with calculus require the creativity of deriving new facts from known facts as part of their solution.  So the study of proof is the study of problem solving.  I don’t require that you have a research mathematician’s sensibility, or that you think proofs are pretty, but you should keep struggling with proofs, as they are an excellent way to improve problem solving and mathematical thinking skills all around.

That being said, the emphasis on proof will decrease somewhat, to make room for other things in lecture.  They will still be required in homework, however.  I’ve found that the art of proof is much more easily taught in office hour, in an interactive environment.  So come visit me and I’ll happily work through some with you, in Socratic method.

“Marking is harsh.”

A couple of students said this.  The marking is a little harsh, agreed, but not wantonly so.  But that’s not a bad thing, necessarily, especially in view of the fact that grades are scaled.  Yes, I’m requiring that you work hard at writing very clear solutions.  This is a higher-level math course, where precision of exposition is expected to a higher degree, and working hard on this skill will bring you great benefits on your exams.  I know you will all rise to the challenge.  (And don’t be discouraged by a 12/15: just think of it as a 4/5!)

“More handing in of homework, less quiz.”

Ah, you don’t trust my dice, eh?  I’ll think about adjusting the weighting, since I agree that we shouldn’t spend too much time on the quiz, and that having homework feedback is important.  It is good to study your homework well enough to be tested on it, though.

“Sometimes I find it hard to hear you when people are coming in late. / Thanks for making people be quiet.”

To latecomers:  don’t come late.  To talkers:  don’t talk.  If you must come late, please talk to me about why (I have had one student do this, and I understand there are sometimes reasons, but we can’t have lots of people doing it).  My father, who teaches university as well, just locks the auditorium doors at the beginning of class.  I haven’t been that harsh, yet….

“The room is like a sauna.”

This has been reported to the appropriate university services.  I didn’t realise:  I guess I thought I was just sweating from the mathematical excitement.

Comments on Computer Graphing

Mixed reviews on the requirement for computer graphing.  Overall, more people find it helpful than find it annoying.  It’s meant to be a small part of the course, and will only be assigned as a small part of homework and no part of testing.

Comments on Homework

Mixed reviews on this.  Some say too long, some say too easy, some say it’s not in synch well enough with lectures.  I’m taking these suggestions into consideration.  In general, I try to assign homework that will prepare you for exams.

“Can we have office hours on Tuesday?”

I wish.  But I have a joint position at UBC and SFU and on Tuesdays I am doing research at SFU, generally.  When possible to have some free time at UBC, I’ll announce an extra office hour.  I’ll make some extra time the week of midterms.

Comments on Pace

Of those who commented, half said slower, half said quicker.  This is probably good, because the pace is dictated by the syllabus, so I can’t do too much more or less than I’m doing.

What do you find hard?

Thanks for answering this; it’s always helpful.  For those who gave specific answers, please come see me in office hour, and I’ll happily spend time going over whatever you’d like.  The general themes of difficulty were: graphing, proofs, curvature, remembering old calculus, and computer graphing.  I’ve helped students with all of these in office hours, which I will happily do more of, and I’ll also try to work some review into lecture as possible.


Last Minute Homework Hints (Hwk 3)

14.3 #12

First, make sure the parametrisation you found doesn’t have square roots in it:  review my wordpress blog post for hwk #1, 14.1 #36, here.  See also example 6 in section 14.1.

What bounds to use for the arclength integral?  In order to answer this, first answer the question “What shape is the curve of intersection?”  The shape has a simple name.  Then, ask “What is the common sense meaning of “arclength” for a shape like that?”  Then decide on the bounds of the integral.

Homework 14.3 #12: don’t bother with the estimate!

Hi all,

I changed my mind; don’t bother with the estimate on 14.3 #12.  Just ask Mathematica to do the integral for you and be done with it.  It’s too big a mess!  (And my apologies for making this decision so late.)  Note: Sage or Wolfram Alpha will do the integral for you, and that’s acceptable to use (just note you used it).

Homework Hints 14.3 #12, #16

Important Announcement: The homework has been slightly modified, in that several of the questions have been delayed to next week, in case I don’t cover as much as I should in class on Monday.  Please check the website.

Here are some hints.

14.3 #12

[EDIT: Don’t do the estimate after all!  Just get to the integral and use technology to estimate it. More here. ]

This is a great review question.  First, it makes you go back to curves of intersection and finding parametrizations.  Then, you try to find the arclength and, if all went according to plan, you’re stuck with an integral that you don’t know how to do.  And that sends you back to the real old days:  how do you estimate a definite integral numerically if you can’t do it by finding a convenient antiderivative?  Hint: think of the definition of an integral.  You did do problems of this sort long ago in the mists of your calculus past!  You are allowed to use a computer or calculator for arithmetic, but I want you to show/explain the calculation you did to get the number (plugging in a “numerical integral of…” command to a computer is not sufficient, although can be useful for checking your answer).  Important:  it says “correct to four decimal places” so you had better explain why your answer has that precision for full marks!  (PS –  the numerical estimate part of this probably isn’t a good choice for quiz question, but the first part could be.)

14.3 #16

This one is a little hairy in the algebra.  I don’t generally like to choose problems with algebra that can trip you up, but at the same time, it’s good to occasionally practice the zen of algebra: think before you leap!  Think about which method (e.g. which differentiation rule to use) will be the least messy.  And to keep you on track, you should find that: a) when you calculate ds/dt, you will take the square root of something which is actually a square; b) the arclength expression s(t) is an integral that you will find listed in the back of your textbook under “Basic Forms” with a simple answer; and c) the arclength parametrization that you get is a very very very familiar one, after you’ve simplified it with enough trigonometric identities.  And you know, this is really what math is like sometimes!  Messy but rewarding of patience.  When you’re done, lean back, consider the original question and the final answer and say to yourself, “wow”, are those really two parametrisations for the same curve?!  (PS – I won’t put this on the quiz if we have one:  that would be cruel.)

Comments on Homework Solutions

Hi everyone,

I’ve told you you must explain your reasoning in presenting solutions to receive full marks, and in this post I hope to clarify by example what I mean by that.  Remember, clear writing demonstrates clear understanding.  So if you’re having trouble writing clear solutions, go back to asking yourself in which part of the solution is your understanding a bit fuzzy?

I’m posting solutions to homework on the course website, but I wanted to draw attention to the fact that the solutions provided with the textbook (which I use for the posted solutions) are very cursory, and would not receive full marks as presented (!).  They are intended only as a guide to the core steps of a problem, not as a finished solution.

As I’ve tried to make clear, the best answer provides a commentary on why each step is part of the path to the solution.  I hope that grading will help you learn what constitutes a good solution.  But I am also going to present some example solutions to the last quiz problem here, and what is involved in getting full marks on such a problem.  By the time we get to the final exam, you will all be experts at writing clear solutions!

Question from the Quiz:

Let \mathbf{u}(t) = \langle u_1(t), \ldots, u_n(t) \rangle be a vector function, and let f(t) be a scalar function.  Prove that

\frac{d}{dt} \left( \mathbf{u}(f(t)) \right) = f'(t) \mathbf{u}'(f(t)).

You may use the following Theorem: Let \mathbf{u}(t) = \langle u_1(t), \ldots, u_n(t) \rangle be a vector function.  Then \mathbf{u}'(t) = \langle u_1'(t), \ldots, u_n'(t) \rangle.

Justify each step in your proof.

Example Solution 1 (80%):

\frac{d}{dt} \left( \mathbf{u}(f(t)) \right)= \left\langle u_1'(f(t))f'(t), \ldots, u_n'(f(t))f'(t) \right\rangle = f'(t) \left\langle u_1'(f(t)), \ldots, u_n'(f(t)) \right\rangle = f'(t) \mathbf{u}'(f(t)).

Commentary: This solution contains nothing false.  That’s really great!  But the reader is left to decipher what happens at each “=” sign, and sometimes two or more things happen at once.  That means it’s hard for the reader to follow.  The instructions clearly said “justify each step” and this solution does not do so.  This is the sort of solution you’ll see in the textbook solution sets: it’s a guide to the core steps and not a complete written solution.  On the quiz, a solution like this got about 12/15, or 80% (footnote:  this is exactly the same as 4/5, believe it or not).

Example Solution 2 (100%):

\begin{array}{lll} \frac{d}{dt} \left( \mathbf{u}(f(t)) \right)&= \left\langle \frac{d}{dt}( {u}_1(f(t))), \ldots, \frac{d}{dt} ({u}_n(f(t))) \right\rangle & \mbox{by the Theorem} \\&= \left\langle {u}_1'(f(t))f'(t), \ldots, {u}_n'(f(t))f'(t) \right\rangle & \mbox{by the Chain Rule for scalar functions} \\&=  f'(t) \left\langle {u}_1'(f(t)), \ldots, {u}_n'(f(t)) \right\rangle & \mbox{pulling out the scalar }f'(t) \\ &=  f'(t) \mathbf{u}'(f(t)) & \mbox{by the Theorem} \end{array}

Commentary: This solution does what was asked: it proceeds from one expression to another, justifying each step and doing only one thing per step.   It is organised on the page.  It receives full marks.

Example Solution 3 (110%):

Let \mathbf{u}(t) = \langle u_1(t), \ldots, u_n(t) \rangle be a vector function, and let f(t) be a scalar function.  The derivative \frac{d}{dt} \left( \mathbf{u}(f(t)) \right) can be calculated by taking the derivative of each coordinate of the vector function \mathbf{u}(f(t)).  This reduces the problem to one about differentiation of scalar functions, where the usual chain rule applies.  The result is a “chain rule” for vector functions.  To be precise,

\begin{array}{lll} \frac{d}{dt} \left( \mathbf{u}(f(t))  \right)&= \left\langle \frac{d}{dt}( {u}_1(f(t))), \ldots,  \frac{d}{dt} ({u}_n(f(t))) \right\rangle & \mbox{by the Theorem}  \\&= \left\langle {u}_1'(f(t))f'(t), \ldots, {u}_n'(f(t))f'(t)  \right\rangle & \mbox{by the Chain Rule for scalar functions}  \\&=  f'(t) \left\langle {u}_1'(f(t)), \ldots, {u}_n'(f(t))  \right\rangle & \mbox{pulling out the scalar }f'(t) \\ &=  f'(t)  \mathbf{u}'(f(t)) & \mbox{by the Theorem} \end{array}

This demonstrates the chain rule for vector functions, i.e.

\frac{d}{dt} \left( \mathbf{u}(f(t)) \right) = f'(t) \mathbf{u}'(f(t)).

Commentary: This solution goes the extra mile in explaining the idea of the proof.  This helps the reader and the writer understand the proof as a gestalt concept, instead of a chain of steps (and it will help the writer study later too).  The idea “vector derivatives are made up of scalar derivatives on each coordinate, so chain rule for scalar gives a chain rule for vector” can be much more easily internalised and remembered than the sequence of steps.  This is what we’re aiming at.  Although this solution is a little wordy, everything it says is relevant, correct, and clarifying, and I would give it bonus points for demonstrating very clear understanding.  However, be warned:  Wordiness is dangerous because the more you say the more likely you are to say something false, which is very bad, or irrelevant, which is somewhat bad.  So please be extremely careful about each sentence that you write:  false or irrelevant wordiness will lower your grade, because this demonstrates lack of understanding.

Example Solution 4 (90%):

Let’s solve a derivative of a vector, which we do by a scalar.  The crux of the matter is the chain rule.  We have a chain rule for vectors because of the scalars, since the derivative of a vector is a scalar.

\begin{array}{lll} \frac{d}{dt} \left( \mathbf{u}(f(t))   \right)&= \left\langle \frac{d}{dt}( {u}_1(f(t))), \ldots,   \frac{d}{dt} ({u}_n(f(t))) \right\rangle & \mbox{by the Theorem}   \\&= \left\langle {u}_1'(f(t))f'(t), \ldots, {u}_n'(f(t))f'(t)   \right\rangle & \mbox{by the Chain Rule for scalar functions}   \\&=  f'(t) \left\langle {u}_1'(f(t)), \ldots, {u}_n'(f(t))   \right\rangle & \mbox{pulling out the scalar }f'(t) \\ &=  f'(t)   \mathbf{u}'(f(t)) & \mbox{by the Theorem} \end{array}

Commentary: This is what I mean by confusing writing.  While presumably this (imaginary) student had in mind the same general idea as the last student did, the student says something which is ambiguous at best and false at worst (“the derivative of a vector is a scalar”), and is generally imprecise in the use of mathematical language (you don’t “solve a derivative” although you can “take a derivative”; and it’s best to say “derivative of a vector function” instead of “derivative of a vector” since you can only take a derivative of a function, not any old vector; and by the way, which scalars does he or she mean by “because of the scalars”?).  To prevent this sort of thing, have someone read what you’ve written (without your help in explaining) and provide some feedback on whether they could understand it.  This solution is correct in the mechanics, but I may remove a few marks for the very confusing preamble, since it demonstrates a lack of understanding.

Computer Graphing Help

I’ve had a few requests for some help with the computer graphing due on this assignment.  I’ve found a few more resources over the past week to help you out, so here are a few links.

Getting Sage Going

The main Sage server has been really laggy, and doesn’t look like it will get any better anytime soon.  You can find some alternative, less laggy Sage servers (you’ll have to apply for a username again) here.  I’ve put up examples of graphing commands here.  Just go ahead and cut and paste them (without the “sage:” part), changing the functions to the ones you want to graph.  You can also install Sage on your own computer, but if you have Windows, be warned that it takes lots of space and requires installing VMWare first.  It is free, however.

Getting Maple Going

Maple can be used in the LSK labs, or you can buy a copy online (the university probably has student rates if you investigate).  Click on “Online Demos” at the course website and you’ll find login information for the LSK labs (be warned: try this at least a day in advance since if you have trouble with your login, you’ll have to email me to get one and it takes time; also labs are closed at certain hours).  When you have Maple working, go here and scroll down to find some graphs of parametric curves.  Go right ahead and cut-and-paste the commands (don’t cut and paste the “>”).  Important: in Maple you have to put the command with(plots): once at the beginning of your session in order that the plotting commands work.  You only need to do this once.

Once you’ve got it going…

Cut and paste an example command, then just change the function to graph another function.  Make sure to test the command before and after changing it to figure out what you’ve done wrong if it stops graphing.  Try changing the parameters.  If you need more help figuring out the meaning of the options, or finding more options (you can change the colour, you can make the line thicker, etc.) two resources are your friends:  the “online demos” section of the course website (where I have lots of internet links) and Google.

Please play around a little, explore!

Please try changing or setting “plot_points” which is default just 75.  This means that in a very curly or complex plot, the computer won’t use enough points to get an accurate picture of the plot.  In sage, you put plot_points=[1000] to use 1000 points.  In maple you add numpoints = 1000.  Try setting these to 5 points to see what happens!  You can also put color=”blue” in Sage and colour=blue in Maple, etc.  For an example of commands that include tages like these in both Sage and Maple, look here.

My purpose in making you do this is not that cutting-and-pasting is valuable work, but to make sure that you each discover the capability of graphing things, so that in later homework you feel that you have this tool in your arsenal to help with understanding.  The computer is a very useful tool in mathematical exploration.  You might find that even in a fairly abstract homework problem, it could help to graph some examples to get rolling.

Math 317 Virtual Office Hours Blog

I’m moving “Virtual Office Hours” onto a wordpress blog for three reasons:

1) Because WordPress supports LaTeX, a typesetting language that makes it easy for me to write math in a post.  For example

\int_a^b sin(t)^2 dt

2) Now you can follow it on RSS so I don’t have to email you every single time something is updated.

3) You can write comments in response to posts, so this can be a bit more of a discussion.  Please do do this!

Last Minute Homework Hints (Hwk 1)

Last Minute Homework Hints

How are you all doing with that homework? I’ve decided to mention a few last minute homework hints (the kind of hints that are good after you’ve already done your homework, but possibly missed some important details)!

The first concerns problem #26, which has two parts (read it carefully!). This hint concerns the first part. If you plug in the given values of x, y, and z into the two equations for the surfaces to see if they satisfy these equations, then you are checking that the parametric curve is *in* the intersection of the two surfaces. But this doesn’t prove that they make up the *entire* intersection. Be sure to address that question! That’s your hint!

The second concerns problem #36, which you should only do after problem #26. First, if your vector function answer has a ‘plus/minus’ square root in it, then it isn’t a function (a function must have a unique value for each input t). Oh no, you say, that’s what my answer looks like! Well, don’t despair, but to try again, think about the lowly unit circle, x^2 + y^2 = 1. We can parametrize it by

\mathbf{r}(t) = \langle \cos t, \sin t \rangle

as t ranges from negative infinity to infinity (or from 0 to 2\pi ). But we could also parametrise it by the *two* functions

\mathbf{r}_1(t) = \langle t, \sqrt{1-t^2} \rangle \mbox{ and } \mathbf{r}_2(t) = \langle t, -\sqrt{1-t^2} \rangle

as t ranges from -1 to 1. If that’s a surprise, try graphing the parametric curve for each of r1(t) and r2(t). Now, after thinking carefully about this circle example, go back to #36 (thinking about #26 could help too). That’s your hint!

How to Write A Solution

Q: I was wondering if, for the homework, it is okay to “indicate” why/what we do each step by showing our calculations and steps, or if you actually want us to write out why/what we are doing.

If you’re just doing algebra simplifications, go right ahead without writing any english in it. However, if you set two quantites equal, substitute a value for a variable, or some other kind of “big move”, then consider saying why (a couple words suffice, like “evaluating derivative at point x=a” or “taking derivative” or “looking for intersection, so both equations hold, and we can substitute one into the other” or “L’Hospital’s rule”). On a graph, if you’ve used some reasoning in deciding how to draw it, write out that reasoning in point form, for example “graph goes up and to the right as t -> infinity because…”. Good math writing is like any kind of writing: it’s an art to convey what you wish to express to a reader.

Nevertheless, we’re not grading the artistic component heavily, at least not as art. Here’s a good litmus test: if the grader can figure out what you’re doing, and it’s correct, you get full marks for it. If the grader gets confused and lost, even if the right answer is in a box at the bottom, it won’t get full marks. The more english you write, the less confused the grader will be and the more likely to be able to find part-marks when you do make an error. There *is* however, an upper limit: if you are writing paragraphs upon paragraphs of english, they will actually obfuscate what is going on. So try to keep it to the point: tell the grader exactly what you’re doing and then do it.

One last thought: it is my experience that clear writing reflects clear understanding. So if you’re having trouble organising or writing a solution, think carefully about the problem itself and see if you can figure out what part of it is a little fuzzy in your head. Explaining the solution to a friend (or showing your written solution) can be a big help at this stage, since that friend can ask questions and indicate confusing points.

Section 14.1, Question 32

Q: The graph of Question 32, in Section 14.1. Why does the sort of exponential-curve at the upper end of my t bound vary with the values I put in for t? That is, if I allowed t to range from -15 to 5, instead of 15, as I have in the graphs below, the long arm coming out of the sin-like wave would move to the 5. Shouldn’t the path be the same at t=5 for the two?

Really great question! Here are the curves that my computer produced using the commands u = var(‘u’); parametric_plot3d( (u, e^u, cos(u)), (u, -15, 5), plot_points=[1000])

-15 < t < 5

-15 < t < 15

So what’s going on? They look as if they have an “arm” which reaches out around 5 in one case and around 15 in the other case. But that can’t make sense, since the only difference between one curve and the other is what range to graph it in (in other words, we’re only changing what piece we’re looking at, we’re not changing the curve!). The answer? The arm reaches out both places… it’s all a matter of scale. Look closely at how the axes are labelled and you’ll see what I mean! That’s exponential growth for you!

The student who sent this question produced these graphs with Maple 11 using the commands f1:=t;f2:=exp(t);f3:=cos(t);
spacecurve([f1, f2,f3], t=-15..5, axes = normal,colour=blue, numpoints = 4000,labels=[“x”,”y”,”z”]);

-15 to 5

-15 to 15

In these Maple 11 graphs, the axis labelling appears to be wrong, making it even more misleading! However, I suspect the software labelled the y axis in different units (it turns out that the label n means n times 10^5 — this just wasn’t very visible in the picture above). So the computer isn’t actually wrong, but this is why you need to know your software, whichever one you use!

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