Vector operator

A vector operator is a differential operator used in vector calculus. Vector operators include:

• Gradient is a vector operator that operates on a scalar field, producing a vector field.
• Divergence is a vector operator that operates on a vector field, producing a scalar field.
• Curl is a vector operator that operates on a vector field, producing a vector field.

Defined in terms of del:

grad

            ≡
            ∇
          
        
        
          
            div
          
          
            
            ≡
            ∇
            ⋅
          
        
        
          
            curl
          
          
            
            ≡
            ∇
            ×
          
        
      
    
  

{\displaystyle {\begin{aligned}\operatorname {grad} &\equiv \nabla \\\operatorname {div} &\equiv \nabla \cdot \\\operatorname {curl} &\equiv \nabla \times \end{aligned}}}

The Laplacian operates on a scalar field, producing a scalar field:

∇

        2
      
    
    ≡
    div
    ⁡
     
    grad
    ≡
    ∇
    ⋅
    ∇
  

{\displaystyle \nabla ^{2}\equiv \operatorname {div} \ \operatorname {grad} \equiv \nabla \cdot \nabla }

Vector operators must always come right before the scalar field or vector field on which they operate, in order to produce a result. E.g.

∇ f

{\displaystyle \nabla f}

yields the gradient of f, but

f ∇

{\displaystyle f\nabla }

is just another vector operator, which is not operating on anything.

A vector operator can operate on another vector operator, to produce a compound vector operator, as seen above in the case of the Laplacian.

• del
• d'Alembert operator

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• H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.