Math 2E Discussion Section 10 (Fall 2005)
Multivariable and Vector Calculus

Solid Angle and Differential Forms Solutions Posted
Posted on December 7, 2005 at 7:37am

I just posted Dr. Pham's solutions to the solid angle and differential form problems. I plan to repost details on the final exam here later today or early tomorrow. Best wishes - Paul

Midterm 2 Solutions Posted
Posted on December 1, 2005 at 3:16pm

I just posted solutions to the second midterm. I hope to have a few more details for your final exam in the next few days. Thanks, study hard, and have a good weekend! - Paul

PS: Don't forget to fill out your student evaluations! You can find them by logging on at www.eee.uci.edu.

Some Additional Tips
Posted on November 18, 2005 at 12:56am

Here are a few tips to consider when you're studying for your exam:

1. DO: When changing from Cartesian to polar coordinates:
   If you have an integral that you have already set up that has a dA = dx dy in it, then dA = r dr dθ. Notice that you could also get this from doing a Jacobian.
2. DO: When changing from Cartesian to cylindrical coordinates:
   If you have an integral with dV = dx dy dz and you are changing to cylindrical coordinates, then dV = r dr dθ dz.
3. DON'T: When your problem is already in terms of r and θ:
   If your problem is already written in terms of r and θ, then you don't need the extra r in front of the dr dθ. The correct dA is already built into the coordinate system. You only use this extra r when translating a problem from Cartesian coordinates to polar or cylindrical coordinates. Here's an example where you wouldn't need the r dr dθ:

   r(r,θ) = (r cos(θ) , r sin(θ) , 14 ),     0 ≤ r ≤ 2,     0 ≤ θ ≤ 2π.

   In this example, we have a disk of radius 2 that is parallel to the xy-plane and is 14 units above this plane. Since the problem is already written in terms of r and θ, there's no need for the extra r in the dA.

1. Flux:
   Flux is the amount of a field that passes through a surface. So, a differential amount of flux is how much of the field passes through a piece of surface area, i.e.,

   F·dS = (F·n) dS,

   and this is what you'd integrate.
2. Work:
   Work is how much a field or force acts along a path. So, a differential amount of work is how much a force acts along a tiny piece of path, i.e., F·dr, and this is what you'd integrate.

1. Polar coordinates:
   In polar coordinates,
   • x = r cos(θ),
   • y = r sin(θ), and
   • dx dy = r dr dθ.
   Here, θ is the angle between the vector r = (x,y,z) and the positive x-axis.
2. Cylindrical coordinates:
   In cylindrical coordinates,
   • x = r cos(θ),
   • y = r sin(θ),
   • z = z, and
   • dx dy dz = r dr dθ dz.
   Here, θ is the angle with the positive x-axis.
3. Spherical coordinates:
   In spherical coordinates,
   • x = ρ sin(φ) cos(θ),
   • y = ρ sin(φ) sin(θ),
   • z = ρ cos(φ), and
   • dx dy dz = ρ² sin(φ) dρ dφ dθ.
   Here, ρ is the distance to the origin, φ is the angle between the vector r and the positive z-axis, and θ is the angle between the (projection of) the vector r (onto the xy-plane) and the positive x-axis. This page gives a good diagram and information on spherical coordinates.

1. sin²θ + cos²θ = 1
2. sin(2θ) = 2 sin(θ)cos(θ)
3. sin²θ = (1/2)( 1 - cos(2θ) )
4. cos²θ = (1/2)( 1 + cos(2θ) )

Quiz 7 Solutions Posted ; News
Posted on November 17, 2005 at 12:09pm

I just posted the solutions to Quiz 7.

In other news, your §17.7 homework is graded. You can drop by my office by 1:00pm this afternoon to pick it up. Lastly, if you have any other old quizzes or homeworks to pick up, please drop by before 1:00pm or email to schedule an appointment to pick those works up.

Best of luck as you prepare for your midterm! - Paul

Some Notational Irritations
Posted on November 10, 2005 at 2:51pm

I noticed when grading the quiz that some of you are not very careful about notation. Not only does this result in "mathematical nonsense," but it also leads to errors in your work. This makes sense: mathematics is a language with its own set of grammatical rules. When you abuse notation, you wind up changing the meaning of what you wrote, which can only lead to errors. Even if your notational errors don't lead to computational errors, you'll be in a position where you aren't properly communicating with the reader, i.e., the grader. (Or in the future, with your colleagues and supervisors.) Using consistent, proper notation is an important part of recording, presenting, and communicating your ideas.

The most common and worst abuse I see is turning a vector into a scalar. For example:

r = ( x , y , z ) = x i + y j + z k ≠ x + y + z.

The expression on the far right is a scalar, even if you put it in parentheses. All the others are vectors. A vector cannot equal a scalar, so this is mathematical nonsense. From now until the end of the quarter, every time I see a vector set equal to a scalar, I will deduct one point. This includes quiz, midterm, and final exam grading.

Similarly, if you're doing a cross product a × b, then both a and b had better be vectors, and the output must be a vector. If you're doing a dot product, a·b, then a and b must be vectors, and the result must be a scalar. If I see an inappropriate use of a cross product or dot product (i.e., the input or output doesn't make sense), then I will deduct one point.

I saw several people make errors in setting up a cross product, such as below:

This doesn't make sense for two reasons. First, determinants are only defined for square matrices, whereas the thing in the determinant above is 2 rows by 3 columns. Second, even if this determinant were well-defined, all the entries of the determinant are scalars, and thus we would expect a scalar output. However, a cross product is supposed to return a vector. This also indicates that there's something wrong with this notation.

In reality, the cross product should be written as:

This error is worth a one point deduction in future work.

Lastly, in general I see the "=" (equal) symbol being used to equate expressions that aren't equal. This is clearly wrong, and I will deduct one point for this.

This may all seem a little harsh (and in truth, it is), but I firmly believe that if you pay more attention to being careful with your notation, you will ultimately make fewer errors and do better on your quizzes and exams. You've reached a point in your educational and professional careers where you can no longer bluff notation, and where notational errors will increasingly lead to computational errors. So, if you find yourself making these (or other) notational errors, it's imperative that you end these bad habits now. That is my goal in explicitly penalizing for these notational abuses. Please think about it in your future work. Thanks - Paul

PS: One hint I can give that might help you to avoid errors is to clearly mark vectors as vectors. For example, use a bold print (such as on this webpage), or better yet, draw an arrow over the vector.

PPS: I updated the scoring on the Quiz 6 solution.

Quiz 6 Solutions Posted
Posted on November 10, 2005 at 11:03am

Hello, class. I just posted your Quiz 6 solutions. Have a great weekend! - Paul

Quiz 5 Solutions Updated
Posted on November 5, 2005 at 11:08am

I just updated the Quiz 5 Solution with an improved alternate solution that uses spherical coordinates. It's a lot easier this way, so check it out! - Paul

Midterm I and Quiz 5 Solutions Posted
Posted on November 4, 2005 at 12:13pm

I just posted Dr. Pham's solutions to Midterm I, as well as my solutions to Quiz 5 below. If you have any questions regarding Midterm I or grading for that exam, you should address them to Dr. Pham. Thanks - Paul

Oops! Quiz 4 Solutions Reposted ; Spooky Halloween Typos
Posted on October 31, 2005 at 12:39pm

Oops! I didn't transfer the updated quiz solutions properly. I fixed the error. Also, note that there's atypo in the first alternate solution. The integrand should read

|r_r × r_θ|dr dθ,

rather than what is currently printed there. I'll try to fix that up later. - Paul

Quiz 4 Solutions Updated
Posted on October 27, 2005 at 2:51pm

I just updated the Quiz 4 solutions. There are now two alternate approaches given. - Paul

Quiz 4 Solutions Posted
Posted on October 27, 2005 at 12:51pm

I just posted the Quiz 4 solutions. - Paul

Update: I was so impressed with one student's alternative view of the parameterization that I'm going to update my solution and post that. It goes very nicely with the hint I gave this morning. (Note that you still had to determine what the radius was.) - Paul

Update2: Another way to view the problem was to think of x = x(y,z) and use the formula for a surface. In that case, you get the square root of 1+(∂x/∂y)² + (∂x/∂z)² in the integrand. From there, it's best to switch to polar coordinates, i.e., y= r cos(θ), z = r sin(θ), dx dy = r dr dθ, etc. I'll update the solution to show this approach, too.

The Difficult Problem from Tuesday
Posted on October 26, 2005 at 12:30pm

If you're recall, we had troubles on problem 42 in §17.6 this past Tuesday, and I assigned it to myself as a homework assignment. :-) I'm happy to report that I have some progress that may be useful for you. First, I did some simple work to visualize the surface in question:

This matches the description we gave on Tuesday. The blue dots give x² + z² = a², and the red dots give the portion of that cylinder contained in x² + y² = a². We seek to find the surface area of this region. Let's use the symmetry to our advantage. First, let's cut the region in half vertically (along z = 0) and then crosswise (along y=0). Then, let us cut the region down the axis (along x=0). This cut region has 1/8^th of the original surface area. Let us parameterize the region.

Notice first that y freely varies between 0 and a for this cut surface. Also, x and z satisfy x² + z² = a². Thus, the surface is given by

r(θ , y ) = (a cos(θ), y, a sin(θ)).

It remains to determine the range of y and θ. Notice first that 0 < y < a. Also, we can see that the range of θ depends upon the value of y. So, we'll want to integrate with respect to θ first, and y second. Now, when y=0, it is clear that θ ranges from 0 to π/2, when y=a, θ ranges from π/2 to π/2, and for intermediate values of y, θ ranges from the intersection of the cylinders to π/2. The intersection is given by

x² + y² = a² = x² + z²,

so y = ± z. In our case, we take the positive root. So, y = a sin(θ), that is, θ = sin^-1(y/a). So, we first integrate with respect to θ from sin^-1(y/a) to π/2, and then we integrate with respect to y from 0 to a. Lastly, the integrand is

|(∂r/∂θ) × (∂r/∂y)| dθ dy.

I'll leave it to you to do these computations; don't forget to multiply your answer by 8 when you're done. My homework is done. :-)

Class Plans for Thursday
Posted on October 18, 2005 at 10:31am

Hello, class. This Thursday, I plan to bring any and all graded work I have to class, which will include (at a minimum) HW1, HW2, Quiz 1, Quiz 2, and Quiz 3. Please make every effort to attend class this Thursday so I can return your work before the Friday midterm. Also, I am following up with the grader on HW 3, so that may also be available on Thursday.

I plan to devote at least 30 minutes to review and questions on Thursday in preparation for your midterm. However, I also feel that it's important to keep pace with the most recent material, so I'll spend at least some time on §17.5 and §17.6.

Please come prepared with any questions you have on materials up to and including §17.4. Thank you! - Paul

Quiz 3 Solutions Posted
Posted on October 13, 2005 at 9:00am

Quiz 3 solutions will be are posted. by noon on Oct. 13, 2005.

One Last Thing ...
Posted on October 12, 2005 at 8:55pm

Of course, it goes without saying that you should always keep an eye out for situations where applying Green's theorem results in a much easier integral.

A Few Things to Think About Before Quiz 3 ;-)
Posted on October 10, 2005 at 2:11pm

First off, the official word is that the quiz will cover sections §17.3 and §17.4. These are the same sections as are due for your homework this Thursday.

I have an interesting thing to think about. Suppose C is a closed curve, and F₁ is conservative. Then what is

∫_C F₁·dr,

and what happens if you add a non-conservative field F₂ to the mix? That is, think about how you might compute

∫_C (F₁ + F₂)·dr.

When you're in a limited-time situation (such as a quiz), you'll want to be able to use any properties of the problem to your advantage. Here's another situation to think of. Suppose that the vector field F is of the form (0,Q), that is, the vector field is purely vertical/has no component in the x-direction. Suppose your path C is a square with corners at (0,0), (1,0), (1,1), and (0,1). Then that path can be thought of as four line segments. Along which of these segments is the line integral automatically zero? If you can identify them, then you can spare yourself some computation.

Now, suppose that you have a problem where you are able to recognize the vector field F as the gradient of a scalar function f: F = ∇ f. Then you should be able to use the fundamental theorm of calculus in such a problem to make your work easier.

Of course, don't forget about the easy tricks, such as the linearity of integrals

∫(f + g)·dr = ∫f·dr + ∫g·dr,

breaking paths up into separate, easier pieces, pulling out constants, and the like.

Lastly, it's important that you recognize when these tricks don't work. Don't try to apply the fundamental theorem of calculus to a non-conservative field. So, a good first step on a problem is to check to see if the field is conservative. Check to see if the path is closed. Check to see if any of the paths are perpendicular to the field. But above all else, make sure you're positive that any tricks you use really do apply.

These are just a few things you might want to keep in mind as you hone your skills with line integrals and prepare for Quiz 3. Best of luck, and I'll see you all tomorrow. - Paul

Quiz 2 Grades Posted ; Some Musing on Teaching Style
Posted on October 10, 2005 at 12:08am

Sorry for the late update, folks. I've been thinking about an email I received about the latest quiz. It seems that it may have been unclear what sections you were to expect on the quiz. I'm working with Dr. Pham to get a better idea of what to expect on quizzes, and how closely the quiz topics correlate with the homework being turned in on the same days. I hope to follow-up with you on those issues within a few days.

On the other hand, it's best to consider the quizzes as a method for keeping pace with the course and new content. In that light, it would be best to study the Monday-Wednesday material for the quiz. Again, I'm checking on this, but in any event, this is a good idea, in that it will help prevent you from falling behind on the course.

I also understand that my linking together the notions of conservative fields from §17.1 and §17.3 may have been confusing. If that was the case, I apologize. I was trying to emphasize the links between the content of those two sections, as well as how some of the content in §17.3 is helpful when considering the topics from §17.1.

In the future, I'll try to make it clear when I'm focusing on one section and when I'm trying to link concepts together in a bigger picture. In mathematics classes, we tend spend a lot of time focusing on tiny pieces and isolated tools. For this reason, I feel it's very important to step pack every now and again to gain a broader perspective on how they link together. That's one aspect that I'll be trying to emphasize in our discussions, but again, when I do so, I'll try to make it clearer.

Lastly, this thought about teaching approach was largely brought on by a student email. (And I greatly appreciated it!) This seems like a good time to reiterate that I value student feedback, as it helps me to better determine what is and isn't working, as well as to confirm hunches. So, if you ever have a comment, question, or concern on how we can make better use of the discussion time and help you to learn as efficiently as possible, please, always feel free to send me an email or drop by during office hours. Thanks! - Paul

Clarification on Quiz 2 Solutions and Some Notes
Posted on October 6, 2005 at 12:23pm

On problem 1, it would also be fine to verify that P_y = Q_x. It is also fine to give a physical example (e.g., gravity), but when doing so, you must state why it is mathematically conservative, i.e., show that the force field is the gradient of a scalar function.

Some of you need to be much more careful with your notation. For example, suppose you have a field (P,Q) = (2xy,x²). Do you see the problems in the following statement?:

P = ∂f/∂y = 2x

For starters, P = 2xy, which is not equal to 2x. Also, if (P,Q) is indeed conservative, P = f_x for some scalar function f, not f_y. What really should be written is:

P_y = 2x (= f_x).

It's clear that differentiation by y was intended, but this isn't what was actually written.

Quiz 2 Solutions Posted
Posted on October 6, 2005 at 9:48am

I just posted Quiz 2. I hope to have the quizzes graded by the end of this morning.

On another matter, I realize that I made a slight error in my board work this morning. I wrote

∫_CF · dr = ∫_a^b F(r(t)) · |r'(t)| dt,

whereas it should have been

∫_CF · dr = ∫_a^b F(r(t)) · r'(t) dt.

Mea culpa and my apologies for any confusion. - Paul

Quiz 1 Grades Posted
Posted on September 29, 2005 at 1:35pm

I just posted your grades to Quiz 1; you can view them at eee.uci.edu. Please note that if you are not officially enrolled in this section, then you will not be able to access your grades for the time being. Please email me if that is the case.

Overall, the work was pretty good. A few of you may have needed to work more examples before taking the quiz. (e.g., calculating gradients and directional derivatives without referring to the text, understanding the implicit differentiation examples, etc.) But generally, as the average was 17.12 (86%) with a standard deviation of 15%, things aren't too bad.

Quiz 1 Solutions Posted
Posted on September 29, 2005 at 10:46am

I just posted the solutions to Quiz 1 below. I hope to have the quiz graded today, but I certainly should be graded by Tuesday. I'll send you an email via eee when the grading is done. - Paul

Syllabus Updated ; Tutoring Center Information
Posted on September 28, 2005 at 8:28am

I just updated the class syllabus to include information on the math tutoring center. Click here to download it. The tutoring center, located in Roland Hall Room 414, is a good place to get additional help on Math 2E beyond class and office hours. If you'd like to work with me during tutoring center hours, I'll be there on Mondays from 8:00am to 9:00am.

Test Post and Welcome
Posted on September 22, 2005 at 1:15pm

Welcome to Math 2E. Vector calculus is an interesting and useful extension of the calculus you learned in Math 2A and 2B. As you already know, the world is a dynamic place that can be modeled by equations that describe how things change and how quickly. It was for this very reason that calculus was (independently) co-invented by Newton and Leibniz over 300 years ago; for Newton in particular, the development of calculus was driven by his interest in physics.

Those of you who have studied physics know that most interesting things involve several different rates of change: changes in the x, y, and z directions, as well as changes in time. In fact, it is the interaction of these rates of change that makes physics interesting and sometimes very surprising; single-variable calculus is insufficient for modeling and understanding such cases. In this class, we will develop and learn to use new tools that you can use in your science and engineering courses to investigate a greater variety of physical phenomena. I find this area very interesting and satisfying, and I hope that you will too as we explore the topics of this course.

On this website, I plan to post homework and quiz solutions, as well as any other helpful resources. I highly recommend that you check here often to stay up-to-date on the class. Updates will always be posted at the top of the page to minimize the need for scrolling, etc. :-)

Once again, welcome, and I wish you success in this new quarter! - Paul