Hi there here is a question that I have wrecking my head for quite sometime I hope that the experts could assist me!
Show that I = $\int_c (x^2y dx + 2xy^2 dy)$ is path dependent in the xy-plane.
I know that I have to use the Test for Analyticity by the Cauchy Riemann Equation where Ux = Vy and Uy = -Vx.
However that method only works via a complex function which I am confused as the above question cannot be converted into a complex function ie. z = x = iy.
Please assist. Thank You!
For the path independence on a simply closed region, one only has to show that the curl is equal to the zero vector.
Here: $\vec{F}=(x^2y,2xy^2,0)^T$, with curl$(\vec{F})=(0,0,2y^2-2xy)^T$. Clearly this vector is not equal to zero. Hence, the integral is pathdependent.