How to prove vector identities WITHOUT using levi civita ?

[darksilence]

Mentor note: Thread moved from homework sections as being a better fit in the math technical section.
Multiplying components of both sides are also off limits.
I am trying to derive vector identities on introduction chapters in various EMT books. For example : (AXB).(CXD) = (A.C)(B.D) - (A.D)(B.C)
After a few hours i noticed B.(CXD) = C.(DXB) and replaced B's with AXB's its Done.
AX(BX(CXD)) was even simpler didnt take any time at all.
I want to do that to
∇. (AXB) = B.(∇xA) - A.(∇xB)
∇x(AxB) = ...
∇(A.B) = ...
∇x(∇xA) = ∇(∇.A) - ∇²A etc

So far last 2 days after solving the first two of them just looking them and hoping to see it. What i should do to improve my ability to see them fast ? (I also have to finish half the book in 2-3 weeks before exam so i am hoping to solve this problem in a few days at most.)

 

[jedishrfu]

Perhaps this lecture will help:

https://www2.ph.ed.ac.uk/~mevans/mp2h/VTF/lecture15.pdf

and this one

http://www.robots.ox.ac.uk/~sjrob/Teaching/Vectors/slides6.pdf

or these

http://www.physics.usu.edu/Wheeler/EM3600/Notes02DerivativesOfVectors.pdf

http://www.physics.usu.edu/Wheeler/EM3600/Notes02DerivativesOfVectors.pdf

Some folks seem to prefer component based proofs and you might have to go with it to be prepared for your test.

 

[darksilence]

Thank you. It wasnt what i wanted at all but some way i didnt imagine they were still helpfull. Instead of working on my weakness i will go on with my strengths.