Divergence and Curl

In this topic:

    (1) Define the del operator and the divergence and curl of a vector field.
    
    (2) If   find and   

    (3) Let be a constant vector field. Show that and
    
    (4) Define the Laplacian operator.
    
    (5) State the basic properties of the divergence and curl.
    

Definition (Del, Divergence, and Curl) Let be a given vector field. The divergence of is defined by and the curl of is defined by where



is the del operator.

Example (Divergence) If   find and   

    Solution. We obtain,



and

    

Example (Constant Vector Field) Let be a constant vector field. Show that and

    Solution. Let for constants and Then


and

Definition (Laplacian Operator) Let define a function with continuous first and second partial derivatives. Then the Laplacian of is   



The equation is called Laplacian's equation, and a function that satisfies it in a region is said to be harmonic on

Proposition (Div-Curl Properties) Let and be vector fields with component functions that have continuous first and second partial derivatives. Then

    (i)

    (ii)
    
    (iii)

    (iv)
    
    (v)
    
    (vi)
    
    (vii)
    
    (viii)
    
    (ix)
    
    (x)
    

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