How to prove vector identities WITHOUT using levi civita ?

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SUMMARY

This discussion focuses on deriving vector identities without using Levi-Civita symbols, specifically the identity (A × B) · (C × D) = (A · C)(B · D) - (A · D)(B · C). The user successfully applied the identity B · (C × D) = C · (D × B) to simplify their derivation process. They also explored identities involving the del operator, such as ∇ · (A × B) and ∇ × (A × B). The user seeks strategies to improve their ability to recognize these identities quickly, especially under time constraints before an upcoming exam.

PREREQUISITES
  • Understanding of vector calculus concepts, including cross products and dot products.
  • Familiarity with the del operator (∇) and its applications in vector calculus.
  • Basic knowledge of electromagnetic theory (EMT) and relevant vector identities.
  • Ability to interpret mathematical proofs and derivations involving vectors.
NEXT STEPS
  • Study the derivations of vector identities in electromagnetic theory textbooks.
  • Practice using the del operator in various vector calculus problems.
  • Review lecture materials on vector calculus, such as those linked in the discussion.
  • Explore component-based proofs to enhance understanding of vector identities.
USEFUL FOR

Students preparing for exams in vector calculus or electromagnetic theory, particularly those looking to strengthen their understanding of vector identities and improve problem-solving speed.

darksilence
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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.)
 
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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.
 

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