Independence of Path

In this topic:

    (1) Define independence of path.
    
    (2) Show that the vector field   



is conservative and evaluate for any piecewise smooth path joining to

    (3) State three equivalences for a vector field to be conservative.
    
    (4) Show that no work is performed when an object moves along a closed path in a connected domain where the force field is conservative.
    
    (5) Summarize strategies for computing line integrals:
    

    In general, the value of a line integral depends on the path of integration, but in certain cases, the line integral will be the same for all paths in a given region with the same initial point and terminal point; and if so the line integral is called independent of path. Additionally, a vector field that is the gradient field for a scalar function on some region is called a conservative vector field on that region. So now recall, for functions of two variables, the analogue of the derivative is the gradient, and the corresponding analogue of the fundamental theorem of calculus is the fundamental theorem for line integrals, which to our advantage, offers us a way to evaluate a line integral of a conservative vector field by simply knowing the values of the scalar (potential) function at the endpoints of the path.

    In general, the value of the line integral depends on the path of integration but in certain cases, the integral will be the same for all paths in a given region with the same initial point and terminal point In this case, we say that the line integral is independent of path in

Definition (Independence of Path) The line integral is independent of path in a region if for any two points and in the line integral along every piecewise smooth curve in from to has the same value.

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



is conservative and evaluate for any piecewise smooth path joining to

    Solution. First we determine We find,  
    






and so the vector field is conservative and we can find the scalar potential function. Now we set out to find with Since we know that



Then, we find and so comparing this with the given   we determine that and so So far we have,



Also since, and comparing this to the given we determine and so is a constant with respect to and Therefore, a scalar potential function is



(taking the constant to be zero). Finally,   

Proposition (Independence of Path) If is a continuous vector field on the open connected set then the following three conditions are either all true or all false:

    (i) is conservative on

    (ii) for every piecewise smooth closed curve in

    (iii) is independent of path within
    

Example (Independence of Path) Show that no work is performed when an object moves along a closed path in a connected domain where the force field is conservative.

    Solution. In such a force field where is a scalar potential of and because the path of motion is closed, it begins and ends at the same point Thus, the work is given by   

    Strategy for Line Integrals: Three ways for evaluating a given line integral , are

    (i)  Parametrize and use the parametrization to convert the line integral into a one-variable integral involving over an interval

    (ii)  Check to see if is conservative; if it is, then find a scalar potential function and use the fundamental theorem of line integrals.

    (iii)  If is conservative, find a convenient path with the same endpoints as and use the fact that since the line integral is independent of path.

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Independence Of Path
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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