1.3 Fundamentals of the Theory of Field
31
⎧
⎪ ⎨
⎪ ⎩
rot x a =
∂a z
∂ y
−
∂a y
∂z
rot y a =
∂a x
∂z
−
∂a z
∂ x
rot z a =
∂a y
∂ x
−
∂a x
∂ y
(1.3.88)
Making use of the Hamiltonian operator, the rotation of the vector a can also be
written as
rot a = ∇ × a =
∂a z
∂ y
−
∂a y
∂z
i +
∂a x
∂z
−
∂a z
∂ x
j +
∂a y
∂ x
−
∂a x
∂ y
k (1.3.89)
Equation (1.3.89) can also be written in the determinant form of easy to remember
∇ × a =
i j k
∂
∂ x
∂
∂ y
∂
∂z
a x a y a z
(1.3.90)
The modulus of rotation is
|rota| = |∇ × a| =
∂a z
∂ y
−
∂a y
∂z
2
+
∂a x
∂z
−
∂a z
∂ x
2
+
∂a y
∂ x
−
∂a x
∂ y
2
(1.3.91)
The basic computing formulae of rotation are
∇ × (ca) = c∇ × a (c is a constant)
(1.3.92)
∇ × (a + b) = ∇ × a + ∇ × b
(1.3.93)
∇ × (ϕa) = ϕ∇ × a + ∇ϕ × a
(1.3.94)
∇ · (a × b) = b · ∇ × a − a · ∇ × b
(1.3.95)
∇ × ∇ϕ = 0
(1.3.96)
∇ · (∇ × a) = 0
(1.3.97)
∇ × (ϕ∇ψ) = ∇ϕ × ∇ψ
(1.3.98)
∇(a · b) = (b · ∇)a + (a · ∇)b + b × ∇ × a + a × ∇ × b
(1.3.99)