It is possible to find paths for wich the following two line integral are equal to zero?
$$\int_C xy^2dx+ydy=0$$
$$\int_C \frac{-ydx+xdy}{x^2+y^2}=0$$
Could somebody give me a path for each that satisfaces or tell me why this isn't possible?
2026-04-24 02:16:29.1776996989
Finding a path for an integrals
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For the first one note that $y\,dy=d\bigl({1\over2}y^2\bigr)$. Therefore we take a path $C$ parallel to the $y$-axis (hence $dx=0$ along $C$) connecting two points $(a,\pm b)$, e.g., the segment from$(0,-1)$ to $(0,1)$.
In the second integral we are integrating $\nabla{\rm arg}$ along $C$; hence the value of the integral is the total increment of the polar angle along $C$. This value is $=0$ for any closed $C$ that does not go around the origin.