For the vector field g? =(yexy+4cos(4x+y))i? +(xexy+cos(4x+y))j? g? =(yexy+4cos(4x+y))i? +(xexy+cos(4x+y))j? , find the line integral of g? g? along the curve cc from the origin along the xx-axis to the point (4,0)(4,0) and then counterclockwise around the circumference of the circle x2+y2=16x2+y2=16 to the point (4/2? ,4//2,4/2).

We're computing the line integral



It looks like the circular part of should be along the circle starting at (4,0) and terminating at .

Because integrating with respect to a parameterization seems like it would be a pain, let's check to see if is a conservative vector field. For this to be the case, if , then is conservative iff .

We have and . The corresponding partial derivatives are




and so the vector field is indeed conservative.

Now, we want to find a function such that . We have



Integrating with respect to yields




Differentiating with respect to gives






and so



Because is conservative, and a potential function exists, the line integral is path-independent and the fundamental theorem of calculus of line integrals applies, so we can evaluate the line integral by evaluating the potential function at the endpoints. We end up with

This is too long-standing now get out

Ithink so 1.you go to your profile and your questions and you'll be able to delete them