The curl of a vector field F as in (1) is given by curlF = (OR hus the curl of a vector field is again a vector field (which is an important difference between divergence and curl). We will learn more about the meaning of curl in the applications to fluid flows below But we have alrcady briefly met another application of curl in class, even if we didn't use the term "curl". In Section 7.21 of the Class Notes we learned how to test if a 3D force field F is conservative, meaning t a potential and that therefore work in this force field is path independent and energy conserving (Section 7.23). Here is the precise mathematical theorem behind the facts discussed in Section 7.21: hat it has Theorem: Let F(z, y, 2) be a force field with continuous partial derivatives which is defined in the entire three-dimensional space and such that curl F = 0. Then the force field is conservative Problem 2: Calculate the curl of the following vector fields (in cach case working on the largest (x, y, z) domain where all components of the field are mathematically defined): (a) F(z, y, z)-e'sin yi + eã§cos yj + zk (b) G(x, y, z) = zzi + zyzj-Pk
