6. Given the vector field \(\vec{F}(x, y, z) = \frac{x}{\sqrt{x^2 + y^2}}\hat{i} + \frac{y}{\sqrt{x^2 + y^2}}\hat{j}\), using graph paper (or being somewhat meticulous), plot the vector that results at each of these points (x, y) in the xy-plane: (1, 0), (0.7071, 0.7071), (0, 1), (-0.7071, 0.7071), (-1, 0), (-0.7071, -0.7071), (0, -1), (0.7071, -0.7071). In other words, evaluate the vector field for each of the points (x, y) listed and be sure to plot the tail of that vector at the point (for instance, (0, 1) should result in a vector that is one unit long, points straight up [along the positive y-axis] and whose tail is at (0, 1)). Does the vector field appear to diverge (run from) any particular point or does it appear to curl around some particular point? Compute the divergence of \(\vec{F}(x, y, z)\), \(\nabla \cdot \vec{F}(x, y, z)\). NOTE: the divergence is a scalar quantity as is all dot products. Compute the curl of \(\vec{F}(x, y, z)\), \(\nabla \times \vec{F}(x, y, z)\). NOTE: the curl is a vector quantity as is all cross products. Now compute its magnitude.
