Conservative vector field?

Question

I don't need the full solution, just give me some hints as much as posible, and can someone tell me how this question is related to conservative vector field? in the following part, j is always a subscript, sum(wj,j=1..n) means to calculate the sum of wj from j=1 to j=n, I hope this writing style can be understood cos no way to type the math notation here.

Consider a closed polygon with vertices,v1,v2,...,vn,vn+1=v1, arranged in the positive direction and let v be a point inside the polygon. Show that
v=(sum(wjvj,j=1..n)/(sum(wj,j=1..n))
where wj=tan( (alpha(j-1))/2 + tan(alpha(j)/2) / modulusof(vj-v),j=1,2,...n and alpha(j) is the angle with |alpha(j)|

angle alpha is the angle at v.
and the tan equations is to take half of the angle first, and then take the tangent of the halved angle. Hope this is more clear.

And I really can't find how this question is related to conservative vector field.

Answer 1

Can't give you a good answer as your equations are ambiguous.

In particular, tan( (alpha(j-1))/2 + tan(alpha(j)/2) looks strange. Did you mean:
tan(alpha(j-1)/2) + tan(alpha(j)/2) or (tan(alpha(j-1)) + tan(alpha(j)))/2

Is alpha the angle at v or at one of the other vertices?

More generally, a conservative vector field is associated with a potential that is dependent only on position, not on the path to that position. Also, such a field is generally linear - so you can just add the potentials due to the influences of the vertices defining the polygon.

In particular, it looks like the sum(WjVj)/sum(Wj) is the weighted mean of the vertices. It is clear that any point inside a polygon can be so represented, but not having worked through the equations, I can't say if the weights are properly computed.

I realize this isn't much. I hope it helps.

Answer 2

Any hints? Just wanna know how can we start this problem..


I only know that v(x,y), v_1(x_1,y_1),.... v_j(x_j,y_j).. But then I don't know how to simplify this expression and perform the riemann sum..

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)| 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)..

Answer 3

OH i know the answer! Is this a question from Project 2 of module MA1507 question 4?