Problem 2Vector field , gradient, dot products and multiplication

Problem 2. Consider the vector field V and scalar field T given by V (x, y, 2) = and T(x, y, 2) = xy2. Imagine V describes the motion of a fluid and 7 describes the temperature. However, for this model, differences in temperature affects this fluid flow and we should consider instead the product of the two fields given by U = TV as the model for fluid motion. (a) (3 pts.) Show that VU = (VT ) . V +T . (V. V ). (b) (3 pts.) Show that V X U = (VT) XV+I * (V XV ) (c) (3 pts.) Set up (but do not compute) the integral that computes the flux of U through the unit square in the xy-plane. (d) (3 pts.) Without computing an integral, is the net (or mangnitude of ) flux of the vector field V through the unit square in the ry-plane greater than, equal to, or less than that of the vector field U. Explain. (e) (3 pts.) If you were instead given some nonzero scalar field f and another nonzero vector field W, is it possible that fW can be conservative if W is non-conservative? Explain or give an example of an f and W where W is conservative but fW is not