Let Then the line integral is
where and When you substitute you get
.
The question
Consider the vector field F(x, y, z) = (yz + 1)i + xzj + (xy + )k.
Evaluate along the path C from (0, 0, 0) to (1, 0, 0), following the helix (x, y, z) = (cos(t), sin(t), t) from (1, 0, 0) to (1, 0, 4 ) and then the straight line from (1, 0, 4 ) to (0, 0, 0).
My attempt
I first found a scalar potential with the property = F, which is
So the vector field is conservative. I noticed that the path is actually a closed loop, so if I'm not mistaken, I can apply the Fundamental Theorem of Line Integrals. Thus, the solution should be 0, right? However my book says -1. :/
Where have I gone wrong? Thanks.