(1) Find the real and positive constants and such that the following velocity field V is conservative V(x, y, z) = [21x sin (772)]; +[√T ₂²e-Y]; + [x² cos(172)_2₂€¯Y] K V. (II) Consider a force field F(x,7,₁²)=√(x₁4₁²) where is the conservative form of from part (1). Find & such that. F- (III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the points on 2-0 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 minimised (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modelling 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. (IV) Find the work done by F in moving a particle along any closed path C. (V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10. Find the work done by F in moving a particle along path C2 from r(O) to r(2). (VI) Consider two paths C3 and C4, described by the parametric equations #] [²²] = α²₁ e-t [cont]. sint 't [sint. cost +aze-t For simplicity, set a₁ = 0 and α²₂=1 (24) [+] = α₁ [cat]. -2 sin (2t) to define the equations for C3 and cy. (g) Find the equation for C4, in cartesian coordinates (x,y). [Check that your equation makes sense by comparing your result to the plot in part (f).] +α2 + [ USE MATLAB to plot (3 and (y from r(0) to r() indicating the direction of increasing (may manually add direction to plot) sin (24) (h) Find the work done by the force field & in moving a particle along path C4, from r(0) tor (Given Gie) - Ezcas (2) E-toin234 (12sinh (tt) MAST20029 Engineering Mathematics Formulae Sheet 1. Change of Variable of Integration in 2D JJ₁ f(x, y) dxdy = [[ f(x(u, v), y(u, v))|J(u, v)| dudv R 2. Transformation to Polar Coordinates x = r cos 0, 3. Change of Variable of Integration in 3D [[[ f(x, y, z) dadydz 4. Transformation to Cylindrical Coordinates x = r cos 0, y = r sin 0, 5. Transformation to Spherical Coordinates 7. Work Integrals y = r sin 0, LFG 8. Surface Integrals x = r cos sin o, y = r sin sin o, z = r cos o, = = [[[_ F(u, v), w)|J(u, v, w) dudvdw [[₁, g(x, y, z) ds = S J(r,0) = 6. Line Integrals Jo f(x, y, 2) ds = ["* f(x(t), y(t), z(t)) √x'(t)² + y(t)}² + 2'(t)² dt F(x, y, z). dr = J(r, 0, z) = r = r cb dx - [² R² +F="/ F₁ F₂ dt a F. ÂdS = J(r, 0, 0) = r² sin o 9. Flux Integrals For a surface with upward unit normal, 11. F + F3 dz = [[ g(x, y, f (x, y)) √[f² + f² +1dxdy dt = SS₁₂ - ² -F₁fx - F2fy + F3 dydx