Calculus 3 Timed Quiz 8

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.

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.

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)
    

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

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

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  _______   ________.

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