Vector Field

Definition (Vector Field) A vector field in is a function   that assigns a vector to each point in its domain. A vector field with domain in has the form   



where the scalar functions and are called the components of A continuous vector field is one whose components are continuous.

Example (Vector Field) Sketch some representations for each of the following vector fields.

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)
    

Definition (Gradient Field) Let be a differentiable function. The vector field obtained by applying the del operator to is called the gradient field of

Definition (Conservative Vector Field) A vector field is said to be conservative in a region if for some scalar function in The function is called a scalar potential of in .

Example (Scalar Potential) If   



then find a function such that

    Solution. If there is such a function, then  



Integrating with respect to   



Then differentiating with respect to we have



and this yields Thus and we have   



Finally, differentiating with respect to and comparing, we obtain and therefore, a constant. The desired function is



with

Definition (Connected Regions) A region in the plane is called connected (one piece) if it has the property:

    (i) any two points in the region can be connected by a piecewise smooth curve lying entirely within
    
and a simply connected region (no holes) is a connected region that has the property:

    (ii) every closed curve in encloses only points that are in
    

Proposition (Conservative Vector Field) Consider the vector field   



where and have continuous first partials in the open, simply connected region in the plane. Then is conservative in if and only if on

Example (Conservative Vector Field) Determine whether or not the vector field  



is conservative.

    Solution. Let and Then since



is not a conservative vector field.

Proposition (Criterion for Conservative Vector Field) Suppose that the vector field and are both continuous in the simply connected region of Then is conservative in if and only if

    Note that a vector field   



in can be regarded as the vector field   



in Since



we have if and only if

Example (Criterion for Conservative Vector Field) Show that the vector field   



is conservative and find the scalar potential function.

    Solution. Since is conservative. Now we set out to find
Since    we set   



Since   



so    and so we set   



Since   



so    and so we set   

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Cite this as:
Vector Field
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-field.html