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# Vector field

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Consider the vector field
F(x,y)= (-yi+xj)/(x^2+y^2)

Question1)Show that F is the gradient of the polar angle function teta(x,y)=arctan(y/x) defined over the right half-plane x>0 .

Question2)Suppose that C is a smooth curve in the right half-plane x>0 joining two points :
A:(x1,y1) and B(x2,y2).Express "integral(F.dr)"on C, in terms of the polar coordinates (r1,teta1) and (r2,teta2) of A and B.

QUESTION3)Compute directly from the definition of the line integrals:
"integral(F.dr)" on C1 where C1 is the upper half of the unit circle running from (1,0) to (-1,0);
and "integral(F.dr)" on C2 where C2 is the lower half of the same unit circle.

QUESTION4)Since F=Grad(teta) at any point of the plane where vector F is defined (not just in the right half plane x>0), th vector field F ought to be conservative (path-independant).
THIS IS TRUE IN SOME REGIONS, but not in others.
a) Give an example of a region in which vector F is conservative, and justify your answer using the fundamental theorem of calculus for line integrals.

b) Give another example of a region in which F is not conservative, and explain why this does not contradict the fundamental theorem.