Problem 1. (max. 15 points) Consider the following partially known vector field on R3: F(x, y, z) = (yz2 - ycos(xy), xz2 - xcos(xy), g(x, y, z) ) for some unknown smooth function g : R3 - R. You are given that (i) F is irrotational, (ii) at the origin, F takes the value F(0, 0, 0) = e3 = (0, 0, 1), (iii) and finally, its divergence div F equals div F(x, y, z) = (V . F) (x, y, z) = (x2 + 2) sin(xy) + 2xy . Find the function g(x, y, z).Remark 1. Let U be an open subset of 3. (a) Recall that a vector field F = (f1, f2, f3) on U is called irrotational if the associated one-form fidx + f2dy + f3dz is closed. Equivalently, it is called irrotational if curl F = V x F = 0, that is, the identically zero vector field. In this case, we also say that F is curl-free. (b) A vector field G = (91, 92, 93) on U is called incompressible or solenoidal (note that both terms refer to the same thing) if the associated two-form gidy Adz + 92dz A dx + 93dx A dy is closed. Equivalently, G is called incompressible (or solenoidal) if div G = V . G=0, that is, the identically zero scalar field. In this case, we also say that G is divergence-free