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Showing posts from March, 2009

### Introducing Cool 12 Second Math Tricks: Logging Up

A big part of making math problems easier is recognizing simplification patterns.
Hence, Cool 12 Second Math Tricks! More detailed steps follow each video

Cool 12second Math Tricks: Logging up on 12seconds.tv

Details
1. The first expression is the derivative of a logarithm. It's just a matter of seeing it.
2. Remember the chain rule vernocular: "The derivative of the whole thing is equal to the derivative of the outside times the derivative of the inside". You can read every little detail here at Wikipedia.

### Coil Levitation with Eddy Currents

I tried out a little quickie experiment in the lab this afternoon. In short: a coil with a changing current, (AC), placed on a non-ferromagnetic conductor, like aluminum, will induce an opposing magnetic field and levitate. You can read all about the effect caused by eddy currents, on Wikipedia, and watch what happened in the lab here:

### 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…