Astron. Astrophys. 356, 301-307 (2000)

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

and

wherever .

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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