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:

[Graphics:Images/index_gr_2.gif]

The spherical polar coordinates  [Graphics:Images/index_gr_3.gif],  [Graphics:Images/index_gr_4.gif],  and  [Graphics:Images/index_gr_5.gif]  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  [Graphics:Images/index_gr_6.gif]  and where  [Graphics:Images/index_gr_7.gif],  [Graphics:Images/index_gr_8.gif],  and  [Graphics:Images/index_gr_9.gif]  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  [Graphics:Images/index_gr_10.gif]  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  [Graphics:Images/index_gr_11.gif],  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