Skip to main content

Understanding Spherical Gradients

Previously we looked at where the 1/r term comes from in the gradient in cylindrical coordinates. This time, we're looking at the gradient for spherical coordinates.



The spherical coordinate values are shown in the figure below. The new theta coordinate is another angular coordinate similar to the phi coordinate introduced in the cylindrical system. It's angle sweeps down from the positive z axis of the Cartesian coordinate system to the negative z axis. Theres a another change. Instead of being anchored on the z axis and moving up and down, the r coordinate is anchored permanently at the coordinate origin.



In this coordinate system, the the angle theta and the radial direction sweep out circles in vertical planes similar to the horizontal plane circles discussed in the cylindrical case. Because of this, the theta coordinate has the same 1/r multiplier discussed in the cylindrical case.

phi and r sweep out circles in horizontal planes exactly as in the cylindrical system. But now, the radius of the circles are not just dependent on the value of r anymore. They also depend on the value of theta. If theta is zero radians, then the circle swept out by r and phi is a point, a circle with 0 radius. If theta is pi/2 radians, then the circle swept out by r and phi actually has a radius of r. The figure below shows a view of the relation ship between r and theta. The phi circles are swept out by the radial line with lenght r and angle theta from the z axis.



Because phi no longer sweeps out circles with radius r, but with radius r sin theta, it's element of length change must be modified in the same manner and we get the length factor assciate with the phi coordinate shown in the gradient above.

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…