### Does Trivial Actually Mean Tedious?

This installment in the ‘It’s Obvious. Not!’ series relates to the second edition of the book “div grad curl and all that” by H.M. Schey, published by W. W. Norton.

Near the end of the example I referenced here, the author of “div grad curl and all that” states that the following integral is ‘trivial’ and results in an answer of 1/6 pi, (specifically, this falls on page 26 of the second edition). As far as I can tell, the solution is more tedious than it is trivial. I’m hoping there really is a trivial solution. If you know it, please add it to the comments below. I’m posting two versions of the ‘tedious’ solution here.

The integral in question:

The author suggests switching to polar coordinates before solving the integral using the following substitutions:

The substitution that’s not mentioned is:

So, now to solve the ‘trivial’ integral, first use the substitutions mentioned above:

Factoring out the -r squared term in square root:

Using the trigonometry identity

we get:

Which is a little more readable as:

To arrive at the answer, first, we need to take the anti-derivative with respect to r and evaluate the integral over the limits of r stated in the book’s example: 0 to 1. Then, we’ll take the anti-derivative over theta and evaluate over the limits 0 to pi/2.

There are two ways to arrive at the anti-derivative with respect to r, the ‘clever’ way and the ‘go through all the steps’ way.

The ‘clever’ way:

Use the integration tables from Wikipedia. There, you’ll see a table of integrals involving:

Be careful here. u, x, and a are just symbolic notations chosen by the authors of the integration table. They have nothing to do with coordinate systems. a is always considered to be a constant, and x is the variable of the function in question. u is used as a substitution to make the tables easier to read. We want the integral:

So, our anti-derivative over (our) r is:

Evaluating over r gives:

The anti-derivative with respect to theta is:

which evaluates to pi/6. The result promised in the ‘trivial’ evaluation of the integral.

The ‘go through all the steps’ way:

Suppose you didn’t have the integration table, or didn’t think of it. Then, you could perform the integration out step-by-step using the ‘substitution rule’.

For our substitution, we’ll chose

Then, using the 'chain rule' of differentiation:

simplifying and multiplying both sides by dr

and

Now, if we substitute u into our original integral, we get

using the value we found for rdr gives

but,

so, we have

The anti-derivative for this is:

just as we found using the tables. Now you can return to the ‘clever’ solution above and proceed with evaluating this anti-derivative over the limits of r and then finding and evaluating the anti-derivative with respect to theta.

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…