ANNOUNCEMENTS
MTH 254H — Spring 2022

6/12/22

Course grades have been submitted, but won't be visible online until Monday. You can determine your course grade from the table below using your raw score, rounding to the nearest integer if necessary.

• 87–100: A
• 81–86: A$-$
• 70–80: B
• 60–69: B$-$
• 50–59: C

To determine your raw score, add $\frac57$ of your midterm score to $\frac54$ of your final score, then subtract the smaller of $\frac57$ of your midterm score and $\frac58$ of your final score, in all cases including extra credit points, if any.

6/11/22

Final scores have been posted in Gradescope (only).

IF your grade were determined only by your final, it would be:

• 67–80: A
• 65–66: AB (too close to call)
• 55–64: B
• 50–54: BC (too close to call)
• 40–49: C
• < 40: F

As with the midterm, this curve was computed without the extra credit points. If you received such points, you may add them in when determining your grade.

If you have any questions about the exam or how it was graded, ask!

6/1/22

As promised, here is a rough guide to the relevant sections in the OpenStax textbook covered since the midterm.

See below for the corresponding information from before the midterm.

• Vectors are covered in Chapter 2. We covered most of the content in §2.1–2.5.
• We did not cover quadric surfaces, so skip §2.6.
• Cylindrical and spherical coordinates are discussed in §2.7 (using different labels than the ones we used in class).
• Vector-valued functions are covered in Chapter 3. We covered most of the content in §3.1–§3.2 and §3.4.
• We did not emphasize arclength, nor discuss curvature, so it's safe to skip §3.3.
• Partial derivatives are covered in Chapter 4.
• We have now covered most of the content in §4.6–4.8 on the gradient and optimization.
• Multiple integrals are covered in Chapter 5.
• We briefly discussed change of variables (§5.7) in class today, but it will not be included on the final.

This section of our textbook describes five different methods for solving constrained optimization problems.

You do not need to be fluent in all five methods, but you should be very comfortable with at least one.

As requested, I have made a copy of the midterm available, by uploading it to Canvas.

You should be able to access this file by selecting Files from the panel at the left.

I will hold my scheduled office hours this Friday (6/3) and next Tuesday (6/7).

I will also be available most of the day on Wednesday (6/8), so don't hesitate to ask for an appointment.

5/27/22

Final timing

The exam will be released on Gradescope at 2 PM on Thursday, 6/9/22.
The exam must be uploaded to Gradescope by 4 PM the same day.
(The exam will not be available as a Canvas assignment, although you can reach Gradescope through Canvas.)

Final guidelines
As on the midterm, the first question will ask you to certify your agreement to the House Rules, either by signing and including a copy of the cover sheet, or by writing "I agree to the House Rules" on your answer sheet, then adding your signature.

• The final will cover material from the entire course, but with an emphasis on material since the midterm.
• The main new topics (roughly 55–65% of the exam) are:
  • vectors & vector functions;
  • gradient;
  • optimization;
• The old material (roughly 35–45%) is described below in the midterm announcement.
• You may bring two 3″×5″ index cards (both sides) of handwritten notes, or the equivalent.
• Other guidelines are as announced below for the midterm.
• Much of next Wednesday's class will be devoted to review. Come prepared to ask questions!

5/26/22

The first question on this week's assignment asks you to redo one of last week's questions using Lagrange multipliers.

• Using the cross product method ($\grad f\times\grad g = \zero$) is equivalent to using an explicit Lagrange multiplier ($\grad f=\lambda\grad g$) and is therefore an acceptable approach on this assignment.
• If you used Lagrange multipliers on last week's assignment (even though we hadn't yet covered it in class...) you may of course resubmit your (correct) solution for full credit.
• However, if you used Lagrange multipliers on last week's assignment, you are encouraged to submit a new solution instead, which does not need to involve Lagrange multipliers (or the cross product).
  In this case, please add a note to your solution stating that you used Lagrange multipliers last week.

5/25/22

Further information about the vector description of lines and planes can be found in this section of the textbook.

A discussion of the principal unit normal vector, including applications to curvature, can be found in this section.

A basic understanding of the position vector, including descriptions of lines and curves, velocity and speed, and arclength are core concepts for this course. The remaining topics, notably acceleration, the normal vector, and curvature, are enrichment topics that are unlikely to be tested, but which may well arise in applications you see later.

5/23/22

First of all, my apologies again for the technological glitches at the start of class today.

If you can believe it, internet connectivity in HALF of our house went out at 9:55 AM due to a router failure...

To return to the example from the start of class, the goal was to optimize $f=xy$ given $g=x^2+y^2=9$.

We showed in class that $\grad f=y\,\Hat{x}+x\,\Hat{y}$ and $\grad g=2x\,\Hat{x}+2y\,\Hat{y}$. Computing the cross product, $\grad f\times\grad g = 2y^2\,\Hat{z}-2x^2\,\Hat{z} = 2(y^2-x^2)\,\Hat{z}$, and setting the RHS equal to zero again yields $y^2=x^2$.

We have therefore discussed four equally valid approaches to this problem:
using substitution, polar coordinates, Lagrange multipliers, or the cross product.

A summary of the properties of the cross product can be found here.

A similar summary for the dot product can be found here.

The determinant approach to computing the cross product can be found in this week's reading, namely here.

An introduction to the determinant can be found here.

5/22/22

Several students have reported that email messages sent from my campus address have been filed in their spam folder...

If you're expecting an email response from me, please do check if it was filed as spam.

5/21/22

There are several ways of approaching each of this week's homework problems.

Use what you know! And think carefully about "what" vs. "where".

5/20/22

We will not use the surfaces any further in class.

Please return them to the Honors College no later than the end of finals week.

Don't forget to sign the checkout list, as per the instructions.

5/17/22

The gradient can also be used in curvilinear coordinates!

See the textbook for further details.

5/16/22

Further discussion of the hill activity can be found in this article as well as in this followup article.

The first article was written by a former MTH 255 TA who is now a math professor.

5/14/22

We will use the surfaces in class on Monday, 5/16/22, and Wednesday, 5/18/22.

We will most likely not use the surfaces again after this week.

5/13/22

Strange but true: The 13th of the month is more likely to be a Friday than any other day of the week!

Give up? Further information is available here.

5/11/22

I have posted a practice problem that is similar to one we did in class.

The posted solution zaps the given function with $d$, but you should also try to solve this problem using the gradient.

This section of the textbook contains an interface to SageMath that computes and displays the gradient along with level curves.

• Click the "Evaluate" button to run the code shown – or type Shift+Enter.
• You can change the function being analyzed by altering line 6 in the first code block.
• Try $f(x,y)=5-3{*}x^2-y^2$, which will give a reasonable approximation to the Hill activity.
• You may also want to change the domain; try $(x,-1.5,-1.5)$ and $(y,-2,2)$.
• Click and drag the first graph!

5/7/22

You might enjoy this old comic strip about vectors.

This comic strip has already made it into some math classes; see this article

5/5/22

We will use the surfaces in class on Monday, 5/9/22.
We will use the surfaces in class both Monday, 5/9/22, and Wednesday, 5/11/22.

5/4/22

The derivation of the Law of Cosines using the dot product, which I mentioned in class today but didn't have time to present, can be found here.

The short version is that if you use the dot product, the Law of Cosines will take care of itself.

The applet I briefly displayed at the end of class can be found here.

If you've never seen the dot product before – or even if you have – take a few minutes to play with this applet and figure out what the different parts represent. Any questions, ask me!

5/3/22

Midterm scores have been posted in Gradescope (only).

We will go over the midterm in class tomorrow.

IF your grade were determined only by your midterm, it would be:

• 61–70: A
• 55–60: AB (too close to call)
• 48–54: B
• 45–47: BC (too close to call)
• 36–44: C
• < 36: F

This curve was computed without the extra credit points. If you received such points, you may add them in when determining your grade. You may want to review the grading policy. Yes, a good final exam score will throw out your midterm score, but a bad final exam score can not be completely eliminated.

(Yes, there will be $\pm$ grades.)

5/2/22

Below are the answers to the midterm questions.
We will go over the exam in class on Wednesday. Full credit requires correct work.

1. (a) $3y^2~$ (b) $-3y\,\sin(3xy)~$ (c) $-3\sin(3xy)-9xy\,\cos(3xy)~$ (d) $7(x^2+x-y)^6(2x+1)$

2. $-18\pi$

3. $63$

4. $0$

5. $7\pi/3$

6. $16/3$

7. (a) $\approx3$–$5^\circ/\hbox{m}$ (b) [many answers possible] (c) [many answers possible]

EC. $3$

5/1/22

The cover page of the midterm has been posted here.

It includes detailed instructions and the House Rules, so you may want to read it before the exam.

4/29/22

As promised, here is a rough guide to the relevant sections in the OpenStax textbook.

• Multiple integrals are covered in Chapter 5. We covered most of the content in §5.1–5.6.
  (Polar coordinates are discussed in §1.3.)
• We did not yet cover change of variables, so skip §5.7.
• Partial derivatives are covered in Chapter 4. The most relevant sections are §4.3 and §4.5, with §4.1 also useful.
• We haven't yet discussed anything in §4.6–4.8, so skip those sections.
• We did not emphasize the content in §4.2 and §4.4, although you might find it helpful to skim them. The latter section does discuss differentials, although from a somewhat different point of view.
  (I would never, ever write "$dx=\Delta x$", but doing so is unlikely to get you into trouble in this course.)

4/27/22

Midterm timing

The exam will be released on Gradescope at 10 AM on Monday, 5/2/22.
The exam must be uploaded to Gradescope by 12 PM (noon) the same day.
(The exam will not be available as a Canvas assignment, although you can reach Gradescope through Canvas.)

Midterm guidelines
The first question will ask you to certify your agreement to the House Rules, either by signing and including a copy of the cover sheet, or by writing "I agree to the House Rules" on your answer sheet, then adding your signature.

• The exam is closed book!
  You may bring one 3″×5″ index card (both sides) of handwritten notes, or the equivalent.
• You may not discuss the exam with anyone other than me.
  I will be available via Zoom to answer questions.
• Please connect to the regular class Zoom session during the exam.
  You do not need to turn on your video.
• Your answers and supporting work should be written on paper you provide.
  You do not need to print the exam, or include it with your answers.
  
• You do not need to copy the questions.
  (Please clearly label which answer goes with which question.)
• The use of OneNote or similar software that allows you to write electronically and/or annotate PDFs is allowed.
• The use of Word, $\LaTeX$, or similar software that allows you to typeset your answers is acceptable.
  However, doing so can be time consuming... Choose wisely.
• No other software is allowed!
  In particular, you may not use technology to visualize or evaluate mathematical expressions.
  Please shut down all other software during the exam, such as email, phone, browsers, etc.
• No extra time is available for the use of software to prepare and/or upload your exam.
• Your exam must be uploaded to Gradescope by the end of the exam period.

4/25/22

Here are some further resources and tips that may be helpful while preparing for the midterm:

• Good practice questions can be found in the sections on multiple integrals in any (multivariable) calculus textbook.
• A good strategy when integrating is to always ask yourself:
  • What are you adding up? (Volume? Chocolate?)
  • How are you chopping? (Draw a line in the region!)
  • What are the limits?
    (Inner limits: from one end of your line to the other; Outer limits: from the first such line to the last.)
• A good problem-solving strategy is to always start by writing down what you know and what you want.

4/22/22

The most common error on the bead problem on HW 3 was not checking What you were adding up.

Most of you got the limits right (Where) but forgot to multiply by the given "density" function, namely $x^2+y^2$.
Chop, Multiply, Add!

4/21/22

We will use the surfaces in class on Monday, 4/25/22, rather than Wednesday as originally proposed.

Doing so should allow more time for review on Wednesday.

4/20/22

The midterm will be Monday, 5/2/22, in class.

• The topics to be covered on the midterm are
  • multiple integration;
  • partial differentiation.
• The exam is closed book, and calculators may not be used.
• You may bring one 3″×5″ index card (both sides) of handwritten notes, or the equivalent.
• Half of Wednesday's class (4/27/22) will be devoted to review. Come prepared to ask questions!

The exam will be lightly proctored via Zoom, so that you can ask questions during the exam.
You will be expected to confirm adherence to the House Rules, which will be posted beforehand.

4/15/22

My apologies for arriving late to today's office hours, especially to those who couldn't wait.

If you want to set up a time to talk with me over the weekend, let me know via email.

If there is enough demand, I will announce a time here, most likely late Sunday afternoon.

4/15/22

You will need your surfaces for class next Wednesday, 4/20/22.

We will also use them one day the following week, most likely Wednesday, 4/27/22 on Monday, 4/25/22.

The midterm is tentatively scheduled for Monday, 5/2/22 (Week 6).

Part of the previous class will be devoted to review.

Further details will be discussed in class and posted here.

Let me know as soon as possible if this date is a problem for you for any reason.

4/14/22

We didn't get to center of mass in class on Wednesday, which you will need for HW 3.

First of all, read this section of the text on center of mass. (Yes, it's linked to the schedule.)

You can think of center of mass as a weighted average, like computing your GPA. You've chopped your grades up by class, and you need to know the grade in each class, multiply by the number of credits for each class, add over all classes, and then divide by the total number of credits. Here, you know the coordinate value on each small region, you need to multiply by the mass of that region, add over all regions, then divide by the total mass.

Yes, you need to repeat this computation for each coordinate – unless you can find an easier way...

4/13/22

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.

4/12/22

We went over cylindrical coordinates in class yesterday, including those parts of the two activities on the schedule page.

We will start tomorrow's class with the same discussion, but for spherical coordinates.
Try to work through the two activities before class. (They're in the first column for Day 05.)

4/9/22

I have heard from a couple of students that they are struggling...

• My assignments do not include routine exercises. You are encouraged to seek out practice problems from another source, such as the OpenStax text used in other sections. We are currently near the beginning of Chapter 5. Feel free to ask for help with these problems during office hours.
• We set up several integrals in class on Wednesday that we didn't actually evaluate, but for which numerical answers were given. Make sure you can evaluate them and get the expected answer! If not, come to office hours.
• Homework represents a very small part of your course grade. It's important to attempt the problems, and to understand their solutions – eventually. But that can also happen when we go over them afterward, or in office hours.
• I will not consider HW 2 to be late so long as it is submitted by the start of Wednesday's class. (Ignore any online messages to the contrary.) If you're having trouble, come to office hours on Tuesday (or make an appointment).

Here are some specific hints for this assignment:

• How would you find the equation of the line connecting two given points? Can you generalize that process to a plane? What do you think the general equation of a plane is? (Look it up if needed.)
• Always draw a diagram showing the region over which you are integrating! Then use what you know to label the boundaries.
• Chop, multiply, add! Chop where? Multiply what?

4/7/22

The first homework assignment (HW 1) has been graded.

Scores (out of 40) should be visible in the Canvas gradebook, but you'll need to go to Gradescope to see my comments.

We will discuss both the assignment and its grading in class on Monday. (Short version: Anything over 30 is fine.)

Feel free to ask me about this and future assignments during office hours.

4/5/22

You do not need to bring your surface and mat to class until further notice.

I will announce both here and in class when you next need to bring these tools to class.
My best guess at the moment is that we will use them next on Monday, April 20.

4/4/22

The text section on contour diagrams has been updated.

Among other changes, you can use Sage on this page to graph your own function of two variables and draw its level curves.

4/2/22

Several of you have not yet completed the short "separate" assignment announced below on 3/27/22...

... and due yesterday.

4/1/22

I plan to hold an unscheduled office hour TODAY from 11:30 AM to 12:30 PM.

Use the Office Hour Zoom link in Canvas, which is also listed in a Canvas announcement.

3/31/22

Canvas appears to be reporting HW 0 scores as "incomplete"...
Short answer: Please ignore the Canvas gradebook; I don't use it.

You may want to take another look at the grading policy. Homework does not contribute very much directly to your course grade, although successfully completing the homework is of course likely to improve your test scores.

3/30/22

Reminder: Please erase your surfaces and mats after every use.

They can become difficult to erase if the ink sits for too long. Thank you!

HW 0 has been corrected and should be available on Gradescope.

Again, this assignment was just a test of the technology, and does not count toward your course grade.

We briefly discussed the Challenge question from the Park activity at the end of class, which should be enough for you to complete the last question on HW 1.

As with all assignments, you may work with your classmates if desired. If you would like assistance in contacting your group members, please let me know.

3/27/22

Surfaces are available for checkout in the Honors College office.

You will need these manipulatives for class on Wednesday, 3/30/22.

A sample homework assignment has been posted.

• As with all assignments, this one is due at the beginning of class.
• Please familiarize yourself with the information posted internally on Canvas, on the syllabus page, and on Gradescope.

As a separate assignment, please send me a short email message by Friday, 4/1/22.

Please include information about your math background, your major(s), and your motivation for taking this course.

2/1/22

Below is some information about how this course will be run.

Overview:

• Class meetings will be held via Zoom.
• Expect a combination of lecture, discussion, and both individual and group problem solving.
• Some "reading" assignments may involve watching short videos of me explaining a particular concept.
  Watch these videos before class, via Canvas.
• All class meetings will be recorded and available afterward to watch online via Canvas.
• My "boardwork" during class will be available afterward via Canvas.
• All assignments will be submitted via Gradescope.

Details:

• General information about getting started with Zoom is available here.
• General information about submitting assignments via Gradescope can be found here.
• Further information can be found on my own information pages for Gradescope and Zoom.
• Each assignment exists in 3 places: on this website, in Gradescope, and on Canvas:
  • The assignment itself can be found (only) on the schedule page.
  • Each assignment has a name, such as "Use Gradescope" or "HW 1".
  • When you have completed the assignment, export or scan it to PDF.
    Please do not take photographs of your work except as a last resort.
  • Upload your PDF to Gradescope, following the instructions here.
  • After grading, your corrected assignment will be available on Gradescope.
  • After grading, your score will be available on Canvas.

Let me know if you have difficulties with any of these steps.

These instructions are likely to evolve...

Make sure you read the note about textbooks, and take a look at the grading policy.

I reserve the right to make small changes to these rules.