I'm having problems with strange BAC-CAB inconsistencies which makes me loose faith in my ability of solving these types of problems. I need someone to check if this is correct or incorrect.

del = d/dx,d/dy,d/dz

a= constant vector=(a,a,a)

(fi) = scalar field

del^2(fi)=0

Question:

Simplify rot(ax grad(fi)

My attempt at solution:

Using einsteins way with eidi of solving, this would take too long to show in plain text( I dont know LaTeX yet ) so for simplification I use classic vector identities to proove it.

rot(ax grad(fi) = del x (ax grad(fi)) ={BAC-CAB}=a(del del(fi)) - del(fi)(dela)={First part is 0 according to previous definition}= -del(fi)(del *a).

The answer in my book:

-del(fi)(a*del)

Discussion:

The difference between (del*a) and (a*del) is huge, esp. asais a constant vector and will be zero for (del*a). I can not wrap my head around the answer of this problem as all I really have to go on is the acual BAC-CAB rule and nothing else. Please explain to me what might be wrong.