In the space ℝ³, the vector

directed from the origin to a point (x, y, z) is the position vector of this point. When the point is variable, r represents a vector field and its length

a scalar field.

The simple formulae

• ∇⋅r = 3
• ∇×r = 0
• ∇r = = r⁰
• ∇ = − = −
• ∇² = 0

are valid, where r⁰ is the unit vector having the direction of r.

If c is a constant vector, U : ℝ³ → ℝ³ a vector function and f: ℝ → ℝ is a twice differentiable function, then the formulae

• ∇(c ⋅r) = c
• ∇⋅ (c ×r) = 0
• (U⋅∇)r = U
• (U×∇)⋅r = 0
• (U×∇)×r = −2U
• ∇f(r) = f′(r) r⁰
• ∇²f(r) = f′′(r)+ f′(r)

hold.

References