Solve any maths problem

Step by step

1. \mathbf{F} = \left\langle - \frac{y}{x^{2} + y^{2}},\ \frac{x}{x^{2} + y^{2}},\ 0 \right\rangle,\quad \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix}

   The curl measures rotation. Expand the determinant one component at a time.

2. \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - \left(0\right) = 0

   The i component: differentiate R with respect to y and Q with respect to z, then subtract.

3. \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 - \left(0\right) = 0

   The j component: differentiate P with respect to z and R with respect to x, then subtract.

4. \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = - \frac{2 x^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{1}{x^{2} + y^{2}} - \left(\frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} - \frac{1}{x^{2} + y^{2}}\right) = 0

   The k component: differentiate Q with respect to x and P with respect to y, then subtract.

5. \nabla\times\mathbf{F} = \left\langle 0,\ 0,\ 0 \right\rangle

   Assemble the three components.

6. \nabla\times\mathbf{F} = \mathbf{0}

   Zero curl: the field is irrotational, and on a simply connected region it is the gradient of a potential.