Skip to main content

Understanding Cylindrical Gradients

We’ve previously looked at how to derive divergence for cylindrical coordinates. If you’re like me though, knowing the rather lengthy derivation won’t help you understand or memorize the resulting formula. So, let’s take a look at why the result makes sense.

The formula for the gradient of a function in cylindrical coordinates is:



Why is the factor of 1/r in the phi term? Remember what question the divergence is asking. We want to find out the amount the function changes vs. a small change in distance along each coordinate’s direction. For coordinates that actually correspond to distances, like x, y, and z of Cartesian coordinates, or r and z of cylindrical coordinates, this is straightforward. The change in the coordinate corresponds to the change in distance along the coordinate.


For coordinates that correspond to angles in the cylindrical coordinate system, there’s an extra twist. The direction of phi always points tangent to a circle centered on the z axis. The small change in distance along the phi or theta coordinate direction is actually a distance measured along the circumference of a circle. The figure below helps me see the issue more clearly. A small change in phi at a radius of 1 means a change of distance of phi times the radius, or 1 phi. At a radius of 2, the same small change in phi results in a change along the circumference of 2 phi.



The distance used in finding the amount the function changes per small change in distance along the phi or theta coordinate is equal to the change in phi times the radius where the change in phi or theta occurs.

So, to properly account for the distance along the phi direction, the divergence has to be written as:

Comments

Popular posts from this blog

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…