Appendix A: components of b and j with respect to B0
Let us prove that there is generally no simple relation between the components of and along and perpendicular to it. In the following calculations for the vector
the terms , , and vanish because of the collinearity of and for a potential or force-free field , and the properties (34a,b). This yields the expressions
which do not vanish in general. Nevertheless, vanishes in the particular case of a potential field , so that relation (A.7) becomes
This proves that and are both perpendicular to (or ), but we cannot conclude they are collinear. Similar relations to (A5)-(A8) hold for instead of .
Let us notice that Eqs. (17c,d) expressing and are solenoidal
imply after decomposition the relations
which actually do not inter-link the components of and .
In conclusion, and should be decomposed geometrically as follows
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000