Exam 3 (Final Exam) — Wednesday 17 December, 2:30-4:30pm, in our usual classroom.

You are allowed to use both sides of two 8.5” x 11” sheets of paper for handwritten notes, the table of basic derivatives and integrals, the equation sheet, and a calculator. No other notes or technology will be allowed for this exam. If you want to use the derivative/integral table, or your own copy of the equation sheet, you need to bring your own copy to the exam.

This exam will be over the material in Topic 5: Vector Calculus. The relevant chapters in the course text are Chapters 16 and 17.

  • Review problems for Chapter 16 have been posted on WebAssign. For Chapter 17, review the daily problem sets on Green’s, Stokes’, and the Divergence Theorem. Also, review Homework 5.
  • The “big picture” (include lists of what you are expected to be able to do on this exam): topic 5 summary. These are pretty dense, but will give you a feeling for what we’ve covered and how it all fits together.
  • Final exam equation sheet. Contact Dr. Ultman if there are additional equations you think should be included on the sheet. The equation sheet will be projected on the screen during class, but you may also bring your own copy.
  • Links to exams from previous semesters can be found at the bottom of this page. The material covered by Exam 1 this semester corresponds to the material covered by Exams 1 and 2 in previous semesters.

Wednesday 10 December


The divergence Theorem.

Monday 8 December


Stokes’ Theorem.

Example from today’s class. Try the last part yourself — computing the flux of the curl of the field through the disk.

This was a lot of heavy typing at the end of a long day, so one point extra credit on the final for the first person to report a mistake.

Friday 5 December


Green’s theorem examples from class.

Curl and Green’s theorem.

Wednesday 3 December


Vector surface integrals (a.k.a. “flux integrals”).

A few announcements as we head into the end of the semester:

  • Skills Test re-takes must be completed by the end of this week (Friday 5 December). Skills Test re-takes will not be given during dead week or finals week.
  • The final exam will be held in our usual classroom on Wednesday 17 December from 2:30-4:30pm.
  • There will be new material taught on Monday and (possibly) Wednesday of dead week, but the last day of class will be open for review for the final exam.
  • Student evaluations are open online. I appreciate constructive feedback, and take it into account when planning future classes. Please take a moment to complete the evaluation.

Monday 1 December

Welcome back.


Surface integrals.

Friday 21 November


Parameterizing surfaces, surface elements, and scalar surface integrals.

Wednesday 19 November


Conservative vector fields, the Fundamental Theorem of Conservative Vector Fields, path independence, and the curl test.

Monday 17 November


Vector line integrals.

Friday 14 November

I’m going to cancel this afternoon’s class. I suspect road conditions will still be making driving hazardous, and since many of the public school districts are closed, some parents need to be home with their children.

My office hours today are cancelled as well.

See you all on Monday.

Update: An opportunity for extra credit. I’ve posted a review of earlier chapters for vector calculus on WebAssign. It must be completed before class on Monday for credit — no extensions on this one. I don’t know yet how the extra credit will be applied, but it will show up somewhere, somehow…

Wednesday 12 November


Scalar and vector line elements, and scalar line integrals.

Great notes on the vector line element: the bridge book.

Monday 10 November


Introduction to vector fields.

Wednesday 5 November

Exam 2 topics to review:

  • Topic 3 – differentiation in several variables (Chapter 14). Level curves; graphs of functions; the gradient, its properties, how it relates to level curves and directional derivatives; tangent planes, linearizations, and approximations; the differential $df$ (or $\Delta f$); the chain rule in one variable and in several variables; critical points and the second derivative test; Lagrange multipliers.
  • Topic 4 – integration in several variables (Chapter 15). Double integrals: regions of integration, changing order of integration, polar coordinates; changing an integral from Cartesian to polar coordinates. Triple integrals in Cartesian, cylindrical, and spherical coordinates. Applications: mass, area, and volume.

Practice problems for Chapter 14 will be posted later this afternoon.

Practice problems for Chapter 14 have been posted.

Please make sure you review the properties of the gradient — how it relates to the rates of change of the function, and how it relates to level curves of the function.

Tuesday 4 November


Idaho voting information.

Monday 3 November


Integrals in cylindrical and spherical coordinates.

And for everyone taking O-Chem:

today’s xkcd.

(It’s worth checking out the original to see the mouse-over text.)

Friday 31 October


Introduction to cylindrical and spherical coordinates.

Wednesday 29 October


Double integrals in polar coordinates.

Monday 27 October


Triple integrals in Cartesian coordinates.

Wednesday 22 October


Double integrals in Cartesian coordinates, continued.

Monday 20 October


Double integrals in Cartesian Coordinates.

Friday 17 October


Optimization subject to constraint: the method of Lagrange multipliers.

Wednesday 15 October


The plan: today, we will continue (and conclude) the discussion of the second derivative test. Any remaining time will be used for questions and open discussion (WebAssign, general concepts, etc). Problem set #18 on WebAssign has been re-scheduled to Friday.

We will cover optimization using the method of Lagrange multipliers on Friday.

We will be one day behind the posted schedule for a bit, but we will catch up again in the next week or two.

Monday 13 October


Friday 10 October


Proving the properties of the gradient using the geometric definition of the dot product. The chain rule along paths.

Wednesday 8 October


The gradient and directional derivatives.

Monday 6 October


Tangent planes, linearizations of differentiable functions and the differential.

Here are notes on the derivation of the equation of the tangent plane.

There will be time at the beginning of next class to write up your thoughts on the first exam. The questions to be thinking about are:

  • What aspect of the exam was the most difficult? Why?
  • What aspect of the exam was the easiest? Why?

Friday 3 October


Higher-order partial derivatives and Clairaut’s theorem. Differentiability, local linearizations, and tangent planes.

Wednesday 1 October


Level curves and contour maps, continued. Partial derivatives.

Monday 29 September


Introduction to functions of two and three variables: domain and range; graphs; level curves; contour maps.

Some ways to graph surfaces and level curves:

  • Surface/Level Curve Grapher (By Barbara Kaskosz at the University of Rhode Island and Doug Ensley at Shippensburg University.) This is an online applet that will let you graph level curves and graphs of functions
  • Graphing apps for pc’s (Disclaimer: These apps have been recommended by students. I have no personal experience with them. As always, exercise caution when downloading applications from the internet.):
  • Graphing app for the iPhone/iPad/iPod touch: Quick Graph (requires iOS 5.1 or later — again, exercise caution when downloading from the internet).
  • If you have a mac laptop or desktop, there’s a utility called “grapher”. This is what I use in class.

Announcement: I want to talk a little bit more about level curves and contour maps on Wednesday. I’m changing the due date on problem set #12 from Wednesday to Friday. There will be another problem set (#13) also due on Friday, so you should still try to finish as much of problem set #12 as you can before class on Wednesday. In particular, you should definitely be able to do problems 1–10 based on what we covered in class today. You may even be able to do the entire assignment — there’s a tutorial in problem 14 that should help.

Monday 22 September


Computing the normal vector on plane curves and the binormal vector. Curvature.

Friday 19 September


The unit tangent, unit normal, and binormal vectors.

I have rescheduled problem set #10 for next Wednesday (9/23). You could try all of the problems, and are definitely ready for #1-4. I would strongly recommend that you wait until after Monday’s class to do problems #5 & 6, since we will learn a faster way to compute $\hat{N}(t)$ for plane curves on Monday.

Wednesday 17 September


Vector and scalar line elements ($d\vec{s}$ and $ds$), and arc length.

A very good introduction to the vector differential $d\vec{s}$ can be found here (it that link doesn’t work, try this one). This is part of Bridge Book Wiki, authored by Tevian Dray and Corinne Manogue. The Bridge Book is a fantastic supplemental text for our course. You can find a table that gives a correspondence between chapters in Rogawski and pages in the Bridge Book Wiki here.

Monday 15 September


(Note: We will cover the last two slides if time allows. The information on them is not strictly crucial to this class, but may be interesting to anyone who has encountered uniform circular motion in another class.)

Derivatives of vector-valued functions. Tangent vectors, velocity, speed, and acceleration.

Announcement: If you are interested in providing one-on-one mentorship to a low-income high school student, helping them navigate the college application and financial aid process, contact boise@striveforcollege.org. This is a volunteer organization. My understanding is that the time commitment is one hour per week, and mentors are particularly needed this week and next.

Friday 12 September


Introduction to vector-valued functions and parameterized planar and space curves.

Wednesday 10 September


Equations of planes in 3-space.

Monday 8 September


The cross product: algebraic and geometric definitions; mnemonics for computing the cross product; the right-hand rule; geometric properties of the cross product — area, normal vectors, and detecting parallel vectors.

Friday 5 September


Vector projections and orthogonal decompositions.

Wednesday 3 September


Introduction to the dot product: algebraic vs. geometric definitions; computing the dot product; the dot product and angles; orthogonality.

Thursday 28 August


$\hat{i}$, $\hat{j}$, $\hat{k}$ basis vectors; magnitude/unit vector representations of vectors; vector parameterizations of lines.


  • A big thank you to everyone who offered suggestions about ways to better use the presentation technology in the classroom. I’ll start using the split screen next week.
  • I spoke with a WebAssign representative about the problem of being forcibly logged out by WebAssign when trying to access homework. He suggested clearing the cookies and cache of the web browser. Apparently, this fixes several WebAssign issues, so it’s worth trying first when you run into problems. Ideally, you should be logged out of WebAssign when you do it.
  • Have a great weekend.

Wednesday 27 August


Introduction to vectors.


  • You can check Blackboard to see if your clicker registered in today’s class.
  • Please fill out the survey for finding days and times for help sessions with Matt.
  • We will begin class on Friday with questions about WebAssign

Monday 25 August


There was a clear preference for daily problem sets to comprise 15% of the course grade, and homework to comprise 5%. The syllabus will be updated to reflect this.

See you on Wednesday

Welcome to Math 275

Welcome to Math 275, section 002, Fall 2014. Our class meets MWF 1:30–2:45pm, Education Building (EDU), Room 110.

In this course, we will see how the mathematics learned in Calculus I and II generalizes to higher-dimensional settings, and develop new tools to deal with new situations.  In particular, we will explore the mathematics of space-curves (one-dimensional objects sitting in two- or three-dimensional space), surfaces (two-dimensional objects) and volumes (three-dimensional objects).

Please take a few minutes to familiarize yourself with the course website, which includes: announcements, course slides, and notes (which will be posted on this home page throughout the course); the syllabus (this is where you can see how grades will be calculated); information about the textbook and WebAssign (the online homework platform); and exam information.

My email address and links to the log-in pages for Blackboard and WebAssign can be found in the sidebar at the top right side (or, on tablets, at the bottom) of each page of this site.

Blackboard will be used to register clickers, host a class discussion board, and post grades (except for homework and daily problem set grades, which will be posted on WebAssign).

Information on registering for WebAssign can be found here. Information on registering clickers can be found here.

Please mark the following dates on your calendar:

  • Exam 1: Friday 9/26, during class.
  • Exam 2: Friday 11/7, during class.
  • Final Exam: date and time TBA, in our regular classroom.
  • Basic Skills Test (first iteration):  Friday 10/24, during class.

Please feel free to contact me via email at shariultman@boisestate.edu with any questions.