### Gravity Probe B Notes: Projecting Vectors via the Dot Product and the Importance of High School Trig

I'm in the process of reading Schiff's Gravity Probe B inception paper[1].  Gravity Probe B was the satellite borne experiment that detected the Earth's gravitomagnetic field, but that's not what I'll be talking about today.  This post is more about a math trick/pattern.  It's a mathematical pattern that comes up pretty frequently in physics, so I figured it was worth a few notes here.  The first picture below shows the equation for the torque on a spinning object due to a spherical source of gravity, (like the Earth), with a bit of its attendant explanation by Schiff.  My notes can be seen to the left:

The cool part I'm going to focus on today is one of the smallest expressions within the rather ginormous equation 3, (also shown in picture 2):

$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r^2$$

I've run into structures like this in the past and it took me awhile to realize what they did.  Likewise for some of my classmates.  The short version of the story is that this operation gives the projection of the omega vector along the r vector's direction.  It's all done with dot products, and consequently, high school trigonometry.  Here's a picture:

We've got an arbitrary omega and r vector.  The cosine operation shown gives the horizontal component of the omega vector with respect to the r vector.  This is just plain old trig.  The vector equation above accomplishes the same thing with a different notation.  First, we write down the dot product as, (in the G+ version, the following four equations can be found in the pictures as well),

$$\left(\vec{\omega}\cdot\vec{r}\right) = |\vec{\omega}||\vec{r}|cos\theta$$

and we're most of the way there.  The issue is that the dot product carries along a factor equal to the length of the r vector.  That's not what we want, so we divide it out giving:

$$\left(\vec{\omega}\cdot\vec{r}\right)/r = |{\omega}|cos\theta$$

and we're done... except we're not.  In the expression from Schiff's paper we have a vector that points in the r direction.  To get this, we can just multiply the above numeric result times the r vector

$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r = |{\omega}|cos\theta\; \vec{r}$$

There's only one last issue left.  The r vector points in the correct direction, but it has the magnitude of r attached to it.  Consequently, when we multiplied by it, we wound up with our result being r times bigger than it should be again.  So... we just divide out another factor of the length of the r vector to get

$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r ^2= |{\omega}|cos\theta\; \hat{r}$$

and we finally have the component of the omega vector pointing in the r direction!  The r vector with a hat instead of an arrow over it indicates that the vector points in the correct direction, but that its length is one.

References:
1.  Schiff's paper describing the original ideas behind Gravity Probe B, the satellite experiment that detected the Earths' gravitomagnetic field
Schiff, L.I., "Possible New Experimental Test of General Relativity Theory", Physical Review Letters, 4, (1960), 215

### 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…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…