Teaching double integrals: why you should *always* sketch the region

The problem (from Larson’s Calculus, an Applied Approach, 10’th edition, Section 7.8, no. 18 in my paperback edition, no. 17 in the e-book edition) does not seem that unusual at a very quick glance:

if you have a hard time reading the image. AND, *if* you just blindly do the formal calculations:

which is what the text has as “the answer.”

But come on. We took a function that was negative in the first quadrant, integrated it entirely in the first quadrant (in standard order) and ended up with a positive number??? I don’t think so!

Indeed, if we perform which is far more believable.

So, we KNOW something is wrong. Now let’s attempt to sketch the region first:

Oops! Note: if we just used the quarter circle boundary we obtain

The 3-dimensional situation: we are limited by the graph of the function, the cylinder and the planes ; the plane is outside of this cylinder. (from here: the red is the graph of

Now think about what the “formal calculation” really calculated and wonder if it was just a coincidence that we got the absolute value of the integral taken over the rectangle

Hitting the bat with the ball….the vector calculus integral theorems….

When I was a small kid, my dad would play baseball with me. He’d pitch the ball and try to hit my bat with the ball so I could think I was actually hitting the ball.

Well, fast forward 50 years to my vector calculus final exam; we are covering the “big integral” theorems.

Yeah, I know; it is but, let’s just say that we aren’t up to differential forms as yet. 🙂

And so I am giving them classical Green’s Theorem, Stokes’ Theorem and Divergence Theorem problems….and everything in sight basically boils down to integrating a constant over a rectangle, box, sphere, ball or disk.

I am hitting their bats with the ball; I wonder how many will notice. 🙂

Poincare Conjecture and Ricci Flow

My area of research, if you can say that I still have an area of research, is geometric topology. Yes, despite everything, I’ve managed to stay moderately active.

One big development in the past decade and a half is the solution to the Poincare Conjecture and the use of Ricci Flow to solve it (Perelman did the proof).

As far as what the Poincare Conjecture is about:

(If you’ve had some algebraic topology: the Poincare Conjecture says that an object that has the same algebraic information as the 3 dimensional sphere IS the three dimensional sphere, topologically speaking).

Now the proof uses Ricci Flow. Yes, to understand what Ricci flow is about, one has to understand differential geometry. BUT it you’ve had some brush with vector calculus (say, the amount that one teaches in a typical “Calculus III” course), one can get some intuition for this concept here.

Watch the video: it is fun. 🙂

Now when you get to the end, here is what is going on: instead of viewing a space (such as, say, the 2-d sphere) as being embedded in a larger space, one can talk about the space as being intrinsic; that is, not “sitting in” some ambient space. Then every point can be assigned some intrinsic curvature, and Ricci flow works in that setting.

Of course, one CAN always find a space to isometrically embed your space in (Nash embedding theorem) and still pretend that the space is embedded somewhere else; some “first course in differential topology” texts do exactly that.

Two Media Articles: topology and vector fields, and political polls

Topology, vector fields and indexes

This first article appeared in the New York Times. It talks about vector fields and topology, and uses finger prints as an example of a foliation derived from the flow of a vector field on a smooth surface.

Here is a figure from the article in which Steven Strogatz discusses the index of a vector field singularity:

You might read the comments too.

Note: the author of the quoted article made a welcome correction:

small point that I finessed in the article, and maybe shouldn’t have: it’s about orientation fields (sometimes called line fields or director fields), not vector fields. Think of the elements as undirected vectors (ie., the ridges don’t have arrows on them). The singularities for orientation fields are different from those for vector fields. You can’t have a triradius in a continuous vector field, for example.

Comment by Steven Strogatz

Our local paper had a nice piece by Brian Gaines on political polls. Of interest to statistics students is the following:

1. Pay little attention to “point estimates.”

Suppose a poll finds that Candidate X leads Y, 52 percent to 48 percent. Those estimates come with a margin of error, usually reported as plus or minus three or four percentage points. It is tempting to ignore this complication, and read 52 to 48 as a small lead, but the appropriate conclusion is “too close to call.”

2. Even taking the margins of error into account does not guarantee accurate estimates.

For example, 52 percent +/- 4 percent represents an interval of 48 to 56 percent. Are we positive that the true percentage planning to vote for X is in that range? No. When we measure the attitudes of millions by contacting only hundreds, there is no escaping uncertainty. Usually, we compute intervals that will be wrong five times out of 100, simply by chance.

Note: a consistent lead of 4 points is significant, but doesn’t mean much for an isolated poll.