31. If F = Mi + Nj + Pk is a differentiable vector field, we define the notation F· V to mean + N- дх ду dz For differentiable vector fields F and F2, verify the following identities. а. х (F X F) 3 (F, V)F — (F - DF, + (V -F)F, - (V•F;)F, b. V(F, · F,) = (F¡ · V)F, + (F, · V)F, + F¡ × (V × F,) + F, x (V X F,)

1. Consider two vector fields: F1(x, y) = -yi+ xj and F2(7) = F. (a) Evaluate F and F, at the given points. (x, y) (1,0) (0, 1) (-1,0) (0, –1) F(r, y) F(1, y) (т, у) (1,1) (-1,1) (-1,–1) (1, –1) F(1, y) F(x, y) On the grids shown below, sketch above vectors of vector fields F (b) and F. F;(x, y) = -yi + xj F2(F) = F.

it says "Use symbolic notation and fractions where needed" and this is my only attempy

4. Assume A is a constant vector field and R = xi + yi + zk. (a) Simplify the expression (R. VR) · Ã+ V ·V × V × ((Ã × R) × (R × Ã)). (b) Assuming A the position vector of a point on the unit sphere centered at the origin, evaluate the above expression at R = A.