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**j040507-taylor: Use of Taylor Series in Field Line Tracing**

I wish to evaluate the utility of the following Taylor series for the tracing of the field lines:

The spherical polar coordinates , , and are expressed above as functions of arc length, *s*, along the field lines. We will begin at a given foot point on the unit sphere, where we will have and where , , and will be the *r,* θ, and φ values of the foot point. (Because of cylindrical symmetry, φ is kept equal to zero for the time being.) Since is the unit tangent vector to the field line, which is a known function of the spherical polar variables *r*, θ, and φ, it seems that we can use the chain rule to calculate all the derivatives in the series. Then, if we choose a small increment of arc length , it seems that the Taylor series above will give us an approximation for a second point on the field line. By repeating the process many times, it seems that a good approximation to the field line can be obtained.

Please note that the source Mathematica notebooks for this date are in the **j040507-graphics** folder.

Converted [in part] by *Mathematica*
May 7, 2004