Solve any maths problem

Step by step

1. \mathbf{F} = \left\langle x y,\ - \frac{y^{2}}{2},\ 0 \right\rangle,\quad \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

   The divergence adds up how fast each component grows along its own axis.

2. \frac{\partial P}{\partial x} = \frac{\partial }{\partial x}\left(x y\right) = y

   Differentiate the x-component with respect to x, holding y and z constant.

3. \frac{\partial Q}{\partial y} = \frac{\partial }{\partial y}\left(- \frac{y^{2}}{2}\right) = - y

   Differentiate the y-component with respect to y, holding x and z constant.

4. \frac{\partial R}{\partial z} = \frac{\partial }{\partial z}\left(0\right) = 0

   Differentiate the z-component with respect to z, holding x and y constant.

5. \nabla\cdot\mathbf{F} = \left(y\right) + \left(- y\right) + \left(0\right) = 0

   Add the three partial derivatives.

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

   Zero divergence everywhere: the field has no sources or sinks. For a velocity field this is incompressible flow.