(For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the de...

(For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the definition of (A · ∇)B.

Refrence: (ii) and (vi).

Reference: Prob. 1.22:

(a) If A and B are two vector functions, what does the expression (A · ∇)B mean?

(That is, what are its x, y, and z components, in terms of the Cartesian components

of A, B, and ∇?)

(b) Compute (ˆr · ∇)ˆr, where ˆr is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate (va · ∇)vb.

Reference: Eq. 1.21:

Prove product rules (i), (iv), and (v).

(i) ∇( f g) = f ∇g + g∇ f,

(iv) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B),