Exercises 6 and 7 on pages 505-6

Hi Vasco

In your attached note, you query the wording of Braden's Theorem because of its use of the word 'streamlines'. You say the field under discussion ".... is not sourceless and so does
not have a stream function and hence no streamlines."

I do not follow this last bit. Why does a streamline have to have a stream function? It certainly does for an analytic field...but does not have to have one for a general field.

To me a streamline is a line whose tangent at any point is in the direction of the field at that point. This is how it is first introduced on the top half of p453 referring to [3]. Now [3] clearly does have a source and has streamlines and well defined steam function. So I take it you mean a field which is sourceless everywhere it is well defined.

In developing ideas in the text he does seem to apply the concept of streamlines to fields in general and then later focuses in on analytic fields. For example Section 4 on p478 (on the geometric view of divergence and curl) applies to a general field and explicitly does so by considering stream lines and equipotentials (see [5] which is also explicitly not sourceless as |X| increases across the dpds element). The streamline based formulae (5) and (6) are general (and are used for non-analytic fields in Ex 1) and it is only in Section 5 that he starts to narrow the focus down to sourceless fields.

So in my reading, streamline is a general concept. The streamlines of divergence free irrotational fields are a subset that have special significance in that they can be associated with well-defined stream functions. But a stream function is not required for a streamline to exist (based on the tangent definition).

On this reading, the wording can stand as is. However, as usual, I may well have missed something important so would welcome any elaboration you could give to your "hence no streamlines".

telemeter