This is what I got (both with vector identities and actually expanding it out)
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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(a x 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(a x grad(fi) = del x (a x grad(fi)) ={BAC-CAB}= a(del del(fi)) - del(fi)(del a)={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. as a is 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.
Okay, so we know that what i'm doing is wrong. Here is what I know about the BAC-CAB rule;
Directly applying this to the original question will yeild my first answer and not the one you provided.
Please tell me why direct computation of the BAC-CAB rule wont work. As I see it it's now the BAC-CBA rule. I've had this problem many times before, and I think I never noticed because I was working around the answer rather than the theory. Please help me understand why I get it wrong pretty much every time. Isn't the rule what is says it is?
EDIT: Lol the LaTeX code worked! First time I used it!
Yes I told you I start to grasp anything that makes sense with the answer rather than the acual theory. Now, please tell me why BAC-CAB does not apply for this case. Why even have a BAC-CAB rule if it does not apply for every case of Ax(BxC)? It makes no sence and it's not like math definitions at all to stray from case to case.
Indeed, that must be the case. That is also why we use einsteins eidi method which I at first thought was stupid as we alredy had an arsenal of laws to prove, as I belived, anything. Thank you, I've got to try and find more on this eidi method so I really understand it all.