ANNOUNCEMENTS
MTH 255H — Winter 2019

3/20/19

Below are the answers to the final.

1. II & IV are divergence free; I & III are curl free
2. (a) not conservative (b) conservative; $y^3+zx^2+ye^z\sin{x}$
3. (a) $0$ (b) $6xy^2+2x^3$ (c) $1$
4. (a) False (b) False
5. $36\pi$
6. $1125\pi$
7. (a) $4r^3\phat-4r^2\zhat$ (b) $0$ (c) $162\pi$
8. $\frac45$

3/15/19

In class today I asked you to determine the flux of $\FF=z^2\,\zhat$ upward through the paraboloid $z=r^2$ for $0\le r\le a$.

The correct answer is $\frac{\pi a^6}{3}$.
You can either integrate over the paraboloid directly, or use the Divergence Theorem to instead find the flux up through the lid ($\pi a^6$) – and subtract the volume integral of the divergence through the inside of the paraboloid ($\frac{2\pi a^6}{3}$).

3/14/19

Happy Einstein's Birthday! And Happy Pi Day!

Extra office hours: I should be in my office most of Monday, 3/18.

I will probably be in my office from 10:30–11:30 AM.
I will definitely be in my office from 1:30–2:30 PM and from 3:30–4:30 PM (and later if there are people still waiting).
I may be in my office at other times, so don't hesitate to check, or to send me a request for an appointment.

3/13/19

Here is the change of variables problem we didn't have time for this morning:

Consider the region $R$ in the $xy$-plane shown in this figure, where $x=\frac12(u^2-v^2)$, $y=uv$.

• Determine the area of $R$.
• Evaluate $\int\limits_R\frac{1}{\sqrt{x^2+y^2}}\>dA$.

You are encouraged to read this section of the book for a discussion of the general case, and the relation to Jacobians.

When reviewing for the exam, the regular textbook (Briggs/Cochran) is a good source of sample problems.

• The "Basic Skills" questions are good practice – do (only) as many as necessary to hone your computational skills.
• Most of the "Further Explorations" and "Applications" are too specific and therefore may not be be helpful. But there are exceptions, such as Problems 26–29 and 37 in §14.7, and Problems 33–35 in §14.8. Use your judgment – or ask me.
• Table 14.4 at the end of §14.8 is a nice summary, although the notation differs slightly from what we have been using.

3/11/19

The final will be Wednesday 3/20/19 from 6–7:50 PM in LInC 345.

• The final will be slightly less than twice as long as the midterm, and will cover material from the entire course (with somewhat more than 50% new material and somewhat less than 50% old.)
• The old material is described below in the midterm announcement.
• The new material emphasizes surface integrals, divergence, curl, and the corresponding theorems.
• This material corresponds to the last two units in the online text.
• You may bring two 3″×5″ index cards (both sides) of handwritten notes, or the equivalent, as well as the handout containing the formulas for divergence and curl in spherical and cylindrical coordinates.
• Other rules are as announced below for the midterm.
• Friday's class will be devoted to review; come prepared to ask questions.

Here are some suggestions for review:

• Make sure you understand each piece of the bathtub problem from the last homework, and preferably multiple ways of approaching each part.
• See if you can set up $d\AA$ (the vector area element) on a paraboloid. For instance, try doing the second question on the Stokes' Theorem lab for a paraboloid, and make sure you get the same answer as for the other integrals in that lab.
• Go over the midterm!
• Seek out problems from a standard text; feel free to consult me for suggestions.

3/10/19

Two mathematicians are talking on the telephone. Both are in the continental United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?

Give up? Some hints can be found here.

3/6/19

Lab writeup for Friday:

Write up a description of your group's work on this week's activity ("Stokes' Theorem").

You should evaluate at least one surface integral explicitly; additional surfaces, and a comparison of your results, is optional but highly encouraged.

In today's activity, is $r$ the cylindrical radial coordinate or the spherical coordinate? As discussed in class, either is OK, but make sure you know which choice you have made. If you used both, make sure you explain why that is allowed.

A copy of the handout from today's class containing the formulas for divergence and curl is available here.

It is also possible to compute $\nabla\times\FF$ for the today's vector field, $\FF=r^3\,\rhat$, in rectangular coordinates. Recall the relations $r\,\rhat=x\,\xhat+y\,\yhat$ and $r\,\phat=-y\,\xhat+x\,\yhat$. Thus, $\FF=r^2(r\,\rhat)=(x^2+y^2)(-y\,\xhat+x\,\yhat)$. You may want to try computing $\nabla\times\FF$ starting from this expression, and check whether your answer agrees with you calculation in class.

3/4/19

As pointed out in class this morning, I strognly urge anyone who didn't do well on HW 6 to attempt both problems again and then discuss them with me.

3/1/19

The images of vector fields I showed in class are available here, and the vector fields from today's handout can be found here.

You may want to consider the vector fields in the first and last column on the handout, asking yourself first whether you expect the divergence to be positive, negative, or zero, then checking by computing the divergence symbolically.

2/28/19

Students in this class might appreciate yesterday's xkcd comic strip.

2/27/19

OSU is closed again this morning.

The homework due today is now due on Friday. The schedule has also been revised.

2/25/19

OSU is closed this morning.

The homework due today is now due on Wednesday.

2/20/19

Lab writeup for Friday:

Write up a description of your group's work on this week's activity ("The Fishing Net").

Ideally, you will be able to evaluate your final integral to obtain an answer that agrees with our class discussion...

2/19/19

Examples of how to chop up the triangle we considered in class yesterday can be found here.

2/15/19

Today's task was to find the flux of $\BB=K\frac{-y\,\xhat+x\,\yhat}{x^2+y^2}$ "to the right" (in the $y$-direction) through the square $y=0$, $0\le z\le2$, $1\le x\le3$, where $K=\frac{\mu_0 I}{2\pi}$ is a constant. The answer is $2K\ln3$.

• Can you get this answer yourself by chopping and adding?
• Can you redo the calculation if the same $2\times2$ square is rotated so that it is along the line $y=x$?

The figure I was unable to show during class today can be found here.

Although this figure refers to the electric field, the idea is the same for the flux of any vector field.

2/11/19

Below are the answers to the midterm; ask me if you'd like to take a look at worked solutions.

1. (a) III (b) I (c) IV (d) II
2. (a) False (b) False
3. $-2$
4. $3/2$
5. $14/3$
6. $5/4+e$
7. (a) & (d) conservative (b) & (c) not conservative
8. $0$

2/11/19

A worked solution to the homework is available here.

(Yes, you can also use $d\rr_1\times d\rr_2$ to find the surface elements.)

2/8/19

Some further information about the quaternions can be found at MathWorld or on Wikipedia, and about the octonions can be found on Wikipedia, on my website, or in our octonions book (which discusses quaternions in Chapter 3).
(Note added: this book is available electronically through the OSU library.)

A link to some pictures I took in 2004 at the Brougham Bridge in Dublin, where Hamilton discovered the quaternion multiplication table in 1843, can be found here.

2/6/19

Lab writeup for Friday:

Write up a description of your group's work on this week's activity ("The Cone").

You are encouraged to include more than one way of determining dA, although one method is sufficient.

2/3/19

The midterm is scheduled for Wednesday, 2/13/19, in class.

• The primary focus of the midterm is on line integrals.
• There may also be questions about surface integrals.
• Both of these topics require material from earlier in the term; everything covered in class is fair game.
• This material corresponds to the first three units in the online text, with an emphasis on Unit 3, as well as introductory material from the first 4 sections of Unit 4.
• The exam is closed book, and calculators may not be used.
• You may bring a 3″×5″ index card (both sides) of handwritten notes;
• Please write your exams in pencil or in black or blue ink.
• Monday's class will be devoted to review — be prepared to ask questions about topics with which you are having difficulty.

2/2/19

There is a possibility that the final will be rescheduled.

The final is currently scheduled for Wednesday, 3/20/19, 6:00–7:50 PM.

Please come to class on Monday prepared to discuss whether other times might be possible.

The final exam schedule for all classes can be found here.

2/1/19

A tool for visualizing 3-dimensional vector fields can be found here.

Similar examples can be found here.

1/30/19

Lab writeup for Friday:

Write up a description of your group's work on this week's activity ("The Wire").

• Your writeup should address Questions 1 and 2, as well as 3ab.
• You do not need to discuss Question 3c, as we have not yet learned about conservative vector fields.
  (You may of course choose to address this question anyway.)
• You do not need to write up today's second activity on Circulation.

Last week's homework asked about circulation without defining it...

Please feel free to resubmit your answer to Question 3 now that you know the definition.

1/29/19

As noticed by one of you on your homework, there is a very nice online tool for graphing vector fields here.

You may need to adjust the scale parameter to get a reasonable graph.

1/28/19

In class today we considered 3 shapes of wafers:

• straight & vertical: $y=\hbox{const}\Longrightarrow dy=0 \Longrightarrow d\rr=dx\,\xhat+dy\,\yhat=dx\,\xhat \Longrightarrow ds=|d\rr|=dx$
• straight but slanted: $y=x\Longrightarrow dy=dx \Longrightarrow d\rr=dx\,\xhat+dy\,\yhat=(\xhat+\yhat)\,dx \Longrightarrow ds^2=d\rr\cdot d\rr=2\,dx^2 \Longrightarrow ds=\sqrt{2}\,dx$
• circular: $x^2+y^2=100/\pi^2 \Longrightarrow x\,dx+y\,dy=0 \Longrightarrow d\rr=dx\,\xhat+dy\,\yhat=(\xhat-\frac{x}{y}\yhat)\,dx \Longrightarrow ds^2=d\rr\cdot d\rr=\frac{r^2}{x^2}dx^2$
  OR: $r^2=100/\pi^2 \Longrightarrow dr=0 \Longrightarrow d\rr=dr\,\rhat+r\,d\phi\,\phat=r\,d\phi\,\phat \Longrightarrow ds=|d\rr|=r\,d\phi$

The total amount of chocolate on each of these wafers is $50\kappa$, $25\sqrt{2}\,\kappa\approx35\kappa$, and $200\kappa/\pi^2\approx20\kappa$, all in grams.

Make sure you are able to set up each of these three integrals, perhaps using the approach outlined above, and that you can evaluate them to obtain these answers (which requires getting the limits right).

1/27/19

The midterm is currently scheduled for Wednesday, 2/13/19, in class.

If you have any concerns about this timing, please let me know immediately.

1/25/19

A copy of the diagram from class today about different representations of work can be found here.

1/24/19

The UHC has asked me to announce that there is now a math tutoring service in the E/H Conference Room in Sackett Hall, Wednesdays from 5–8 PM, starting this week, and running through Week 10.

Be warned that the tutor is unlikely to be familiar with the geometric reasoning skills I emphasize in this class, notably including the use of $d\rr$.

1/23/19

Lab writeup for Friday:

Write up a description of your group's work on this week's activities ("The Valley" and "Vector Line Integrals").

• The Valley part of your writeup should include a discussion of Question 3 for the path your group was assigned.
• By all means include your evaluations of indefinite integrals if your group approached the problem that way initially, but make sure to (also) include a computation involving definite integrals.
• It should be clear what method you actually used to compute a final answer, although you are encouraged to discuss alternative methods also.
• You are also encouraged to speculate (with or without computing anything) about the answer for path I.
• The Vector Line Integral part of your writeup should include a discussion of Questions 1 and 2.
• Your writeup should clearly indicate what path you chose, and what vector field you were assigned.
• You are encouraged to repeat Questions 1 and 2 for a different vector field.
  If your group was assigned one of vector fields $\{\FF_1,\FF_2,\FF_3\}$, try one of $\{\FF_4,\FF_6\}$, or vice versa.
  You are not required to turn your work on the second vector field, but be prepared to discuss it in class on Friday.

Photographs of today's summary boards can be found here: Path II; Path III; Path IV; Path V.
Copies of the vector fields are posted here.

The vector field images are scans; original PDFs will be posted (and this note removed) when I track them down.

1/16/19

You do not need to write up today's activities ("The Park", "The Hillside", and "The Hill").

However, if you took any photos of your work today, please do send me a copy.

1/15/19

You can find out more about the reasons we will use the "physics" convention for the names of the spherical coordinates in our paper:

Spherical Coordinates, Tevian Dray and Corinne A. Manogue, College Math. J. 34, 168–169 (2003)

The short answer is that most students will need to switch conventions at some point during their education, so this might as well be done sooner rather than later.

1/14/19

With perfect timing, today's xkcd cartoon is about magnetic declination!

You do not need to write up today's activity ("Calculating Line Elements").

Here is a summary of the suggestions from class about writing style and format:

• Reminder: Please write only on one side of a page. (Otherwise, scanning is more difficult.)
• Request: Please avoid paper smaller than 8½×11; avoid pages torn out of notebooks. (Both risk jamming the copier.)
• Provide complete URLs when citing online resources.
• Consider attaching a printout of online figures that you reference.
• Incorporate the question into your answer, so that the reader knows what the question is.
• "This" should almost never standalone; this what?

Make sure you know and understand these formulas in rectangular and polar coordinates: \begin{align} \rr &= x\,\xhat+y\,\yhat=r\,\rhat+0\,\phat \\ d\rr &= dx\,\xhat+dy\,\yhat=dr\,\rhat+r\,d\phi\,\phat \end{align} After today's activity, you should also know the analogous formulas in cylindrical and spherical coordinates.
(The expressions for $d\rr$ can be found here.)

1/13/19

The Briggs/Cochran textbook currently used in regular sections of calculus is on reserve in the Valley Library.

You are strongly encouraged to refer to this text (or any standard calculus text) on a regular basis as a source of practice problems. See me if you are having difficulty choosing appropriate problems.

1/10/19

The schedule has been updated, showing the scheduled final and tentative midterm dates.

A list of derivative rules in differential notation is available here.

1/9/19

Lab writeup for Friday:

Write up a short summary of your work on this week's activity ("Which Way is North?").

• See the homework page for guidance on how I expect written work to be done.
• All written work should be turned in at the beginning of class on the due date.
• You do not need to include descriptive text with your answers to the first two questions, but of course may do so.
• You do need to provide well-reasoned answers to the last two questions.
• You can use this website to determine the magnetic deviation (angle between true north and magnetic north) for any location. You will need to know the latitude and longitude — or the zip code. (Enter a zip code, press the button labeled "Get Location", then press the button labeled "Compute".)
• You can find out more about magnetic declination at Wikipedia, and there are some online maps available here.

Some of the above websites are affected by the current shutdown of the federal government and may not be available.
As an alternative, I have posted several maps online: World; North America; US.

1/6/19

Make sure you read the note about textbooks.

My standard grading scheme is outlined here. I reserve the right to make small changes to these rules.

In this Honors section, group activities may occur any day of the week.

The criteria I will use to evaluate written work can be found here.

Please read the guidelines on the homework page, which also apply to the writeups for the group activities.

A rough schedule for MTH 255 can be found here. Please use this as a guide only.

This schedule is automatically generated, and assumes Thursday recitations, which we don't have.

Recommended readings are listed on the homework page, and will also be listed on the schedule when available.

Feel free to supplement these readings with other content from the Bridge Book, and/or from any (vector) calculus text you are comfortable with.

An online copy of my (slightly outdated) Study Guide for MTH 255 can be found here; a PDF version is available here.

The Study Guide provides a somewhat more traditional treatment of the material we will cover than the approach used in class, which more closely reflects our online book.

You are encouraged to browse the website of the closely related Vector Calculus Bridge Project.