Use the Levi Civit a symbol to prove that (a) (A B) (C D) (A C)(B D) (A D)(B C) (b) (f g) g ( f) f ( g) (c) (A B) (C D) (A C D)B (B C D)A (d) The 2 2 Pauli matrices x , y , and z used in quantum mechanics satisfy i j ij i ijk k If a and b are ordinary vectors, prove that ( a)( b) a b i (a b)

Question: Use the Levi-Civit`a symbol to prove that (a) (A B) (C D) = (A C)(B D) (A D)(B

Use the Levi-Civit`a symbol to prove that
(a) (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C).

(b) ∇ · (f × g) = g · (∇ ×f) − f · (∇ ×g).

(d) The 2 × 2 Pauli matrices σ_x, σ_y , and σ_z used in quantum mechanics satisfy σ_iσ_j = δ_ij + iε_ijk^σk . If a and b are ordinary vectors, prove that (σ · a)(σ · b) = a · b + iσ · (a × b).

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