5/7: Divergence Theorem

Slides: Divergence and the divergence theorem (sec 17.3)

Course grades on Blackboard are now up to date. In addition, I’ve added columns for clicker totals (with lowest three scores dropped), and for homework and exams entered as percentages rather than scores. You can find your current WebAssign score (with the lowest two scores to date dropped) on WebAssign. The rubric for computing the course grade is given on the course syllabus. Please contact me before the final exam if you think there is a mistake in any of your posted scores.

Aaaand, today’s xkcd.

5/4: Green’s & Stokes’ Theorems

Slides: Green’s & Stokes’ Theorems (sec 17.1, 17.2)

No more problems will be added to HW #5. Problem 2 can be simplified (or complicated) by using trig identities, so there is probably more than one way of doing it. Here’s what worked for me.  There may be slicker ways of setting it up; there are certainly messier ways. I would say that the “moral” of this problem is: you can use polar coordinates $x = r\cos\theta$, $y = r\sin\theta$ to parameterize a curve in terms of the angle $\theta$, if you can write your radius as a function of $\theta$.

5/2: Vector Surface Integrals (a.k.a. Flux Integrals)

Slides: Vector Surface Integrals (a.k.a. Flux Integrals) (sec 16.5)

I’ve posted notes on one way to approach HW #5 problem 9. This approach does not use parameterization or change-of-coordinates until after the dot product $\vec{E}\, \cdot \, d\vec{S}$ is computed. If you prefer to try the problem by parameterizing first, then computing the dot product, by all means try it.

I will have information about the final and course grades so far posted early next week. For now, note that the final is almost entirely about the material in chapters 16 & 17 (although I reserve the right to ask basic questions from earlier in the semester — for example, computing dot and cross products, tangent vectors, etc).

Also, the scores posted on Blackboard are not weighted. They are only there so you can make sure I have the correct scores entered in my spreadsheet.

4/30: Surface Elements & Scalar Surface Integrals

Slides: Surface elements and scalar surface integrals (sec 16.4)

4/28: Conservative Vector Fields

Slides: Conservative vector fields (sec 16.3)

Since we didn’t get to the examples of using the curl test, I’ve posted notes on the use of the curl test (there’s also now a link to these notes on the final slide).

Anyone interested on the necessity of the condition that the domain be simply connected in order to apply the curl test: look at example 8 and the following “conceptual insight” on pp 970–971.

4/25: Vector Line Integrals

Slides: Vector line integrals (sec 16.2)


  • It was brought to my attention at the end of class that there is a “point of view” consideration when using the work interpretation of a line integral. The line integral $\int_C \vec{F}\, \cdot \, d\vec{s}$ represents the work done by the field $\vec{F}$. The work done against the field $\vec{F}$ is $-\int_C \vec{F} \, \cdot \, d\vec{s}$: note the opposite sign. I suspect that many people taking physics or engineering classes will tend to think about work done against a field rather than work done by the field; if so, be aware of the difference.
  • Follow-up on the $c(t)$ vs $\mathbf{c}(t)$ notation for parameterized curves: I checked the textbook (sec 14.5 p820). The parameterization is defined to be a “moving point”, and uses curved brackets when writing it out in component form; but it is denoted in boldface $\mathbf{c}(t)$, as though it were a vector. (IMHO, this is a confusing choice on the part of the author/editor — but they may have their own justification for why it makes sense.)
  • Also, a repeat announcement: If you are interested in being a Learning Assistant or Peer Leader, check out the Academic Advising and Enhancement website.

2/23: Scalar Line Integrals

Slides: Scalar line integrals, and the scalar and vector line elements $ds$ and $d\vec{s}$ (sec 16.2)

If you are interested in being a Learning Assistant or Peer Leader, check out the Academic Advising and Enhancement website.

4/21: Introduction to Vector Fields

Slides: Introduction to vector fields (sec 16.1)

We’re in the final stretch. Topic 5 (Vector Calculus) is the final topic we study, and utilizes many of the techniques and concepts previously covered in this course. WebAssign #25 provides a review of material we will be drawing on from chapters 12 through 14. We will also be using multiple integration and polar, cylindrical, and spherical coordinates.

4/16: Solutions

  • HW #4, problem #1 This problem is no longer counted towards the assignment, but I’m including the solution for those of you who have worked on it. The answer and solution that WebAssign will display after the assignment is due, is incorrect.
  • Worksheet II Cylindrical and spherical coordinate solutions.

4/15: Exam 4 Information Posted

Information regarding exam 4 has been posted on the “exams” page of the course website.  Please let me know if you have any questions about what has been posted.  Tomorrow (Wed 4/16) is review.  Come prepared with questions, including any questions you may have about WebAssign and/or homework problems; problems from the worksheets from class on Friday 4/11 and Monday 4/14; and/or the table of coordinate functions and basic curves and surfaces in polar, cylindrical, and spherical coordinates.  Links to the worksheets and the table can be found on “exams” page.

4/9: Polar, Cylindrical, and Spherical Integrals

Slides: Integrating in polar, cylindrical, and spherical coordinates (sec 15.4).


4/8: Final Exam Time Change

I must have misread the final exam schedule at the Registrar’s office when originally scheduling the exam. In order to avoid conflicts and align with the schedule posted at the Registrar’s office, the time of the final exam for Math 275, sec 002, has been changed to 9:30–11:30am on Monday 12 May.

4/7: Introduction to Cylindrical & Spherical Coordinates

Slides: Introduction to cylindrical and spherical coordinates (sec 12.7, 15.4)

Including the volume element $dV$ in cylindrical and spherical coordinates!

4/4: Polar & Cylindrical Coordinates

Slides: Examples of finding limits of integration for triple integrals in Cartesian coordinates (sec 15.3); Introduction to polar and cylindrical coordinates (sec 11.3, 15.4)

(We will begin with cylindrical coordinates on Monday.)

4/2: Applications of Double Integrals; Triple Integrals

Slides: Some applications of double integrals (sec 15.2, 15.5), and triple integrals (sec 15.3)

3/31: Double Integrals in Cartesian Coordinates

Slides: Double integrals in Cartesian (a.k.a. rectangular) coordinates (sec 15.1, 15.2)

3/17: Lagrange Multipliers

Slides: Optimization subject to constraint — the method of Lagrange multipliers

I’ve added a few notes to the end of the last slide (the example of the rectangle inscribed in the ellipse).

3/14: Local Extrema, Saddle Points, and the Second Derivative Test

Slides: Local maxima, local minima, saddle points, and the second derivative test for functions of two variables (sec 14.7).

Here’s a link to a free graphing app for the iPhone/iPad/iPod touch (requires iOS 5.1 or later). I haven’t tried it, but it’s been recommended by a member of our class, and it does look pretty cool.

3/12: Directional Derivatives & Chain Rules

Slides: Directional derivatives & chain rules (sec 14.5, 14.6)

We did not get to the last three slides, but I am posting them anyway. The third-to-last slide is an example of an “applied” problem that uses the chain rule along paths. It uses the chain rule to find how distance from a fixed point is changing with respect to time as one travels along a parameterized curve. You can try it as an exercise: I’ll post a solution separately soon. Solutions to the ice-skating ant and partial derivatives slides.

The last two slides deal with the general form of the chain rule. For the purposes of this class, the general chain rule will be dealt with purely as a matter of mechanics. The “written homework assignment” for section 14.6 is that you read the beginning of section 14.6 (pp 825–828; up to but not including “Implicit Differentiation”), and complete the partial derivative problems on WebAssign #16 (problems #12–14). This material is fair game for exam 3.

3/10: the Differential, the Gradient, and the Directional Derivatives

Slides: The differential, the gradient, and directional derivatives (secs 14.4, 14.5)

We did not get to directional derivatives today. That’s where we will begin class on Wednesday 3/12. Problems on WebAssign #15 involving the directional derivative have been set to zero (they will appear on the next assignment), but make sure you do the problems involving (only) the gradient.

A note on the exam: if you were marked down a point in problem #5 for using “dt” instead of a dummy variable in the integral, bring

3/5: Differentiability, Tangent Planes, Local Linearizations, and Differentials

Slides: Differentiability, Tangent Planes, Linear Approximations, & the Function Differential $df$ (sec 14.4)

We didn’t make it to differentials today: that’s where we’ll start on Monday. The slides have been updated to reflect the material covered.

3/5: Partial Derivatives

Slides: Partial Derivatives (sec 14.3)
(The link is now fixed.)

Here’s another pc graphing app, recommended by a classmate: Microsoft Mathematics. As always, take appropriate precautions when downloading software from the internet.

Announcement: I’ve had to cancel my office hours today. Apologies to anyone who wanted to meet.

3/3: Introduction to Multivariate Functions

Slides: Introduction to Functions of More Than One Variable (sec 15.1)

Summary: Introduction to functions of two and three (independent) variables; domain & range; graphs and level curves of functions of two variables.

There are links on the course Links page to surface and level curve graphers, and to a pc graphing application (disclaimer: I have not used this application. As with anything on the internet, use caution when downloading). If you use a mac, you should have an app called “grapher” in your utilities folder (this is the grapher I use in class). Also, a fun link to online topographic maps.

2/24: Motion Along Curves

Slides: The $\{\hat{T},\hat{N},\hat{B}\}$ frame, and the orthogonal decomposition of acceleration (sec 13.4, 13.5)

Summary: The unit vectors $\hat{T}(t)$, $\hat{N}(t)$, and $\hat{B}(t)$ form a frame of reference for an object moving along a curve. The orthogonal decomposition of acceleration with respect to velocity can be written in terms of the vectors $\hat{T}(t)$ and $\hat{N}(t)$, where the tangential and normal components depend only on speed and curvature.


  • Exam 2 is this Friday (2/28) in class. Information, including a topic summary and a first draft of the equation sheet, is posted on the “Exams” page.
  • Recommendation: have the equation sheet, the derivatives/integrals sheet, and a scientific calculator handy while you’re studying for the exam.
  • Wednesday will be review. I’ll stay as long as people have questions.
  • WebAssign assignment #11 is longer than usual — many of the problems are computationally intensive. It will be due on Friday before the exam. Please read the instructions carefully — it will save you a substantial amount of time (and will very likely help on the exam). You can expect at least one (if not more) question on tangential and normal components of acceleration on the exam. I really like all of the acceleration decompositions problems a lot. You will need to understand how the components relate to speed and curvature. You should take this as a hint.
  • Good on-line curve graphers can be found here (2d), here (3d), and here (3d with tangent and normal vectors). Macs come with a pretty good graphing utility called “grapher” (I’m not sure about PC’s — any suggestions?).

2/21: Unit Tangent Vector and Curvature

Slides: The arc length function (sec 13.3, continued), the unit tangent vector, & curvature (sec 13.4)

Summary: Using the arc length function to find distance travelled along the curve at a given time, and to find the time at which a given distance has been travelled; the unit tangent vector $\hat{T}$; curvature $\kappa$.


  • I’m granting an extension on the extra credit for exam 1. You have until the beginning of class on Friday 2/21 to email me. See the post for 2/19 for details.
  • Homework 2 is due at the beginning of class tomorrow. Remember to follow the formatting guidelines; please fill out your name on both sides of the cover sheet.
  • Grading of this assignment will be slightly different from the first assignment. I will only grade two problems in detail; you choose one, I choose one. The remaining problems with be worth up to two points each, based on the completeness of the attempt. Put a big star next to the problem you would like me to look at in detail. If you don’t choose one, I’ll choose for you.
  • If you are still having problems logging in to the ebook from WebAssign, you can try to log in directly instead. The link is posted under the “Logins” menu on the right sidebar.

2/19: Arc Length Integrals and the Line Element $ds$.

Slides: Arc length integrals and the line element $ds$ (and an introduction to the vector differential $d\vec{s}$) (sec 13.3)

Summary: review of definite integrals — how to “read” and integral; using vector parameterization and definite integrals to compute the length of a curve (arc length); defining and computing the line element $ds$; brief introduction to the vector differential $d\vec{s}$ (aka $d\vec{r}).

Note: the line element $ds$ and the vector differential $d\vec{s}$ are very useful tools we will be using in this course. They are not explicitly described in the course text, but an excellent introduction can be found at the Bridge Book wiki. Note that the Bridge Book denotes the vector differential $d\vec{r}$ instead of $d\vec{s}$.

Announcement: If you want 3 points extra credit on the first exam, send me the following in an email by 5:00pm today (Wednesday 2/19):

  • One factor that was out of your control and had a negative impact on your exam performance.
  • Two factors that were within your control, that would have improved your performance on the exam.

I will compile and post everyone’s answers (without names).

2/14: Derivatives of Vector-Valued Functions

Slides: Calculus along curves (sec 13.3)

Summary: derivatives of vector-valued functions; vector derivative as tangent vector; velocity, speed and acceleration along parameterized curves.

2/12: Introduction to Curves

Slides: Introduction to curves in $\mathbb{R}^2$ and $\mathbb{R}^3$ (sec 13.1)

Summary: vector-valued functions; parameterized curves.

In chapter 13, we will look at functions $\vec{r}(t)$ whose input is a scalar and whose output is a vector (examples of vector-valued functions). The components of a vector-valued function, called coordinate functions, give the coordinates of the terminal points of $\vec{r}(t)$. The curve described by these points is said to be parameterized by the function $\vec{r}(t)$.

2/5: Planes in 3-Space

Slides: Equations of planes in $\mathbb{R}^3$ (sec 12.5)

Summary: similarities between vector equations of lines and planes; derivation of the algebraic equation of a plane using the dot product; normal vectors and their manifestation in the algebraic equation of planes; parallel planes; normal lines; the cross product and normal vectors.

A few announcements:

    • The first exam is on Monday 2/10 during class. More information can be found on the Exams page.
    • Friday 2/7 will be a review day. I will not have anything prepared, but will answer (almost) any question asked during the review. If there are questions that you know you in advance that you would like to discuss during the review, you can send me an email before class.
    • My office hours this semester will be Mondays and Wednesdays from 10:30-12:30 in MG222D. I will also have office hours this Friday (2/7) from 10:30-12:30.
    • The times and locations for Kyle Beserra’s help sessions are:
      • M 4:30-5:45 B222
      • Tu 11-12:15 (room TBA)
      • W 4:30-5:45 ILC 301
      • Th 11:45-12:45 B302
      • F 3-4:45 E221

      (these have also been posted on the course syllabus).

2/3: The Cross Product

Slides: The Cross Product (sec 12.4)

Summary: algebraic and geometric definitions of the cross product; computing the cross product of two vectors; algebraic and geometric properties of the cross product; the right-hand rule.

Watching the polling responses during class, it seems to me that people began getting confused at the geometric definition of the cross product. Keep in mind the following:

      1. We generally do not use the geometric definition of the cross product to actually compute the cross product*, since finding the angle $\theta$ and the unit vector $\mathbf{\hat{n}}$ is not generally possible (same holds true for the dot product — if you are not given the angle as part of the problem, you can’t use the geometric definition there, either).The general rule regarding when to use algebraic and when to use geometric definitions is:
        • Use algebraic definitions to compute.
        • Use geometric definitions to get information about the information content of the computation (what your computation means), or to get a feel for what your algebraic computation should look like using sketches or diagrams.

        There are some questions on WebAssign where you are asked to compute the cross or dot product using the geometric definition, but these are mostly to help you learn the definitions.

      2. The most important geometric information you need to know about the cross product is:
        • The magnitude of the cross product is the area of the parallelogram spanned by the original vectors (this comes from the scalar part of the geometric definition).
        • The cross product is orthogonal to the original vectors, and hence, (if the original vectors are non-zero and are not parallel) is normal to the plane spanned by the original vectors.
        • The direction of the cross product is determined via the right-hand rule.

* There are rare special cases where you can compute using the geometric definition (for example, you can show that i x j = k).

1/31: The Dot Product, Part II

Slides:  The Dot Product Part 2 — Vector Projections & Orthogonal Decompositions (sec 12.3)

Summary: projecting one vector along another (non-zero) vector; scalar components of vector projections; orthogonal decompositions of one vector with respect to another (non-zero) vector.

Given two vectors $\vec{v}$ and $\vec{u}$ with $\vec{u}\neq\vec{0}$, the vector $\vec{v}$ can be written as the sum of two vectors $\vec{v}_{\|}$ and $\vec{v}_{\perp}$, where $\vec{v}_{\|}$ and $\vec{v}_{\perp}$ are orthogonal — that is, $\vec{v} = \vec{v}_{\|} + \vec{v}_{\perp}$ and $\vec{v}_{\|} \cdot \vec{v}_{\perp} = 0$. We will see projections and orthogonal decompositions again in chapters 13 and 16.

1/29: Introduction to the Dot Product

Slides: The Dot Product Part I — Definitions and Angles (sec 12.3)

Summary: algebraic and geometric definitions of the dot product; computing the dot product; using the dot product to compute angles; orthogonality.

The dot product will play a major role in our class. Today we saw that the dot product encodes geometric information about length (distance) and angles.

1/27: Vector Parameterizations of Lines

Slides: Parallel Vectors & Vector Parameterizations of Lines (sec 12.2)

Summary: definition of parallel vectors; vector parameterizations of lines in the plane and in 3-space; similarities between vector parameterizations of lines and the point-slope equation of a line in the plane; coordinate functions (aka parametric equations).

In the larger picture of this class, today’s topic (parameterizing lines) is an introduction to the more general topic of parameterizing curves in the plane and in 3-space. Intuitively, a curve is one dimensional (you can only go forward or backwards along the curve), so you should be able to express the curve in terms of a single variable or parameter.

In previous math classes, we learned that the graph of a function $y = mx + b$ is a line in the plane with slope $m$ and $y$-intercept $b$. In calc III, the roles of slope and $y$-intercept are taken by vectors.

Using vector parameterizations, we can answer questions like: What location in space corresponds to a given parameter value? Does a particular point lie on a line? Do two lines intersect?


      • Today is the last day to drop classes without a “W”.
      • Clicker scores will begin counting for credit on Wednesday 1/27. If you have been using your clicker in class, you can check the “clicker tests” on Blackboard to see if it is working properly.

1/24: Introduction to Vectors

Slides: An Introduction to Vectors (sec 12.1, 12,2)

Summary: an introductions to vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$; vector algebra (scalar multiplication and vector addition); magnitude; unit vectors; and the unit basis vectors $\hat\imath$, $\hat\jmath$, and $\hat{k}$.

Announcement: I’ve added a page Commenting on the Course Website, with instructions on logging in to comment and changing your publicly displayed name. See the links under “Getting Started” on the right sidebar.

1/23: WebAssign Announcement

The “first” assignment (besides the integration/differentiation self-assessments) has been posted on WebAssign. This assignment is due before class on Monday 1/27. I’m adding one extra attempt per problem on this one, so that people who have not used WebAssign before have a little more leeway with learning how to use it.

General Note For This Class: DRAW PICTURES!!!!! (Yes, I’m shouting. You should also imagine me jumping up and down. It’s important.)

Also, I will be opening discussion forums on WebAssign for each assignment. You can find the link to these forums below the list of assignments on WebAssign. Add topics as you see fit. I’ll be following the discussions, but mostly staying out of them; this is a space for you to talk about the assignments with each other.

1/22: General Class Business

Slides: Welcome to Math 275

Summary:  A tour of the course website; going over the course syllabus; registering for WebAssign; registering “clickers” on Blackboard.

On Friday, we will begin talking about vectors.  Something to think about: have you seen vectors before?  What are they?

Welcome to Math 275

Welcome to Math 275, section 002, Spring 2014. Our class meets MWF 9:00–10:15am in the Micron Business & Economics Building (MBEB) room 1210

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).

For now, please take a few minutes to familiarize yourself with the website.  You can get to the Math 275 homepage from any page on my website using either the main menu, or the “Math 275 Home” link in the upper-right hand corner.  If you want to contact me, my email address is on the bottom left of every page.

Information on the course texts can be found here.  Information on registering for WebAssign can be found here. Information on “clickers” (student polling devices) can be found here.  Information about exam dates, grading policies, etc., can be found on the course syllabus.

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