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$ \nabla \times F = \begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\-x & y & 0 \end{vmatrix} = 0 $

Therefore, it is path independent integral and can be integrated along path y = 0 and x = y also.

For path y = 0, dy = 0

$ \int _{1} ^{0} (-xdx + 0) = \dfrac{1}{2} $

For path x = y , dx = dy

$\int _{0} ^{\frac{1}{\sqrt{2}}} (-xdx +xdx ) = 0 $

Add both path integrals to get $\frac{1}{2}$

Therefore, it is path independent integral and can be integrated along path y = 0 and x = y also.

For path y = 0, dy = 0

$ \int _{1} ^{0} (-xdx + 0) = \dfrac{1}{2} $

For path x = y , dx = dy

$\int _{0} ^{\frac{1}{\sqrt{2}}} (-xdx +xdx ) = 0 $

Add both path integrals to get $\frac{1}{2}$