(a) Find the real and positive constants and such that the following velocity field V is conservative V(x,y,z)=[2xsin(z)]i+[z2ey]j+[x2cos(z)2zey]k (b) Consider a force field F(x,y,z)=V(x,y,z) where V is the conservative form of V from part (a). Find such that F=. (c) Can the divergence of F be zero on the plane z=0 ? Justify your answer using the divergence of F on this plane. Classify the points on z=0 as source, sink or neither. No structure operates in perfect isolation. Structures always interact with their environment and these interactions entail energy losses which need to be minimized (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modeling of components in renewable technologies concerns measures of the energy expended in the flows (e.g., air) around them. The work done is one such measure. (d) Find the work done by F in moving a particle along any closed path Q. (e) Consider two paths Q ) and Q. Suppose the work done by the particle in moving through F along path Q ) from S(0) to S(2) is 10 . Find the work done by F in moving a particle along path Q from S(0) to S(2). (f) Consider two paths Q+ and Q, described by the parametric equations in your NOTES on pages 2.25 and 2.27, respectively (go to the "general solution" to the system of ordinary differential equations that are given on each page).* For simplicity, set V=0 and V!=1 to define the equations for Q ) and Q. For example, on page 2.25, the path Q+ would be parameterized by r(t)=[etsin(t)]i+[etcos(t)]j+[0]k Use MATLAB to plot C3 and C4 from r(0) to r(2) indicating the direction of increasing t (you may manually add this direction in your plot). (g) Find the equation for Q, in cartesian coordinates (x,y). [Check that your equation makes sense by comparing your result to the plot in part (f).] (h) Find the work done by the force field G in moving a particle along path C4 from r(0) to r(2) given G(t)=[2tcos(2t)]i+[tsin2t]j+[t3+2sinh(2t+1)]k