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Vector field

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.

Solution Summary

This solution provides a vector field and shows that it is the gradient of a given polar angle function, then gives characteristics of a smooth curve and shows how to express the integral in polar coordinates.
It also shows how to compute integrals using the definition of line integrals, and how to give regions where a vector is conservative and not conservative.