Conservative Vector Field??

Discussion in 'Undergraduate Math' started by SGod88, Nov 10, 2007.

  1. SGod88

    SGod88 Guest

    I am encountering a problem in line integral.. I dont know how to
    relate this to a conservative vector field.. It is good that some
    hints will be given to solve this question..

    Consider a closed polygon with vertices,v_1,v_2,...,v_n,v_(n+1)=v_1,
    arranged in the positive direction (anticlockwise) and let v be a
    point inside the polygon.


    Show that
    sum( w_j*v_j, j=2..n)
    v= --------------------------------------
    sum( w_j, j=2..n)


    where
    tan[ (alpha_(j-1)/2 ] + tan [ alpha(j)/2 ]
    wj= ------------------------------------------------------------
    ||vj-
    v||
    j=2,3,...n +1


    and


    alpha_ j is the angle at v with |alpha(j)|<PI, of the oriented
    triangle [v,vj,vj+1] taking positive value if the orientation is
    positive and negative value otherwise.


    I am not clear how this problem lead to the concept of line integral
    and conservtive vector field...


    So far I found that it only can be written as


    v_j
    sum(w_j*v_j)= ------------------------- * {(tan[ (alpha_(j-1)/2 ] +
    tan [ alpha(j)/2 ] }
    ||vj-v||


    and v=(x,y,z), v_j=(x_j,y_j,z_j)..


    then how should I continue this? Any hints? Thanks a lot!
     
    SGod88, Nov 10, 2007
    #1
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