This exercise concerns the function phi = phi (x, y) = (x^2 + y^2) e^1-x^2-y^2 and its gradient vector field f = nabla phi. see the plots of each below. (a) compute the partial derivatives phi_x and phi_y to find nabla_phi. (b) in ma231, you learned that mixed second-order partial derivatives of reasonable functions should agree. verify that here by computing phi_xy and phi_yx and checking that they are the same. (c) the plots should convince you that this problem exhibits strong "radial symmetry" about the origin, to see this algebraically, express phi and nabla_phi in polar coordinates (r, theta). (d) find the magnitude of the gradient field |nabla_phi| and express your answer in polar coordinates. where is the magnitude zero? (bonus: where is |nabla_phi| maximum? it's doable, but a little messy to do by hand.) (e) without doing any significant computations, what is the work done by the vector field f = nabla_phi counter-clockwise around a circular path centered at the origin? justify your answer. (f) directly compute the work done by f = nabla_phi along the line segment c starting at the origin o(0, 0) and ending at the point p(2, 0) by integrating integral_c f middot t ds using change of variables for line integrals. you may use a computer or a table to do the integral, but show all other work including how you parameterize the path and apply the change of variables. (g) the field f = nabla_phi is a conservative field. use the fundamental theorem of line integrals to find the work done by f along the line segment c and confirm your answer in the previous part.