### Use the Calculus Luke!

Yeah for Google+!!!  After posting yesterday's missive about figuring out the origin of the secant squared term, mathematician extraordinare and Google+er +John Baez pointed out to me that the derivative of tangent is secant squared and that I need not have fussed so much with the geometry.  John said:
I wouldn't  call the introduction of sec^2(θ) a "substitution".  Introducing θ in the first place counts as a substitution, since you're trying to simplify an integral by replacing some other variable with this one.   But it's just a mathematical fact that the derivative of tanθ is sec^2(θ), so
d tan(θ) = sec^2(θ) dθ
I would use this equation automatically and unthinkingly, but it looks like you're trying to find a geometrical explanation of this equation.  If you're figuring it out for yourself because nobody told you the derivative of tan(θ) is sec^2(θ), that's very laudable!  The standard approach is to express tan in terms of sin and cos, and use the quotient rule for derivatives.﻿

I guess the lesson is when you see differential on both sides of an expression, think calculus, not geometry!

### 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…