position vector (original) (raw)

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 , r→ a vector field and its

a scalar .

The

• •
  ∇⋅r→= 3
• •
  ∇×r→=0→
• •
  ∇⁡r=r→r=r→ 0
• •
  ∇⁡1r=-r→r3=-r→ 0r2
• •
  ∇2⁡1r= 0

are valid, where r→ 0 is the unit vector having the direction of r→.

If c→ is a 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→=-2⁢U→
• •
  ∇⁡f⁢(r)=f′⁢(r)⁢r→ 0
• •
  ∇2⁡f⁢(r)=f′′⁢(r)+2r⁢f′⁢(r)

hold.

References

• 1 K. Väisälä: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).