You should be able to:
Compute integrals of vector fields over curves (work integrals) in both vector and differential form notation
Determine if a vector field is conservative.
Find the potential function of a conservative vector field.
Use the fundamental theorem of line integrals to compute work integrals.
Carefully discuss the relationships between various properties of vector fields: being conservative, satisfying the path independence property, integrating to zero around closed loops, having their curve be zero, having a non-simply-connected domain.
Use green's theorem to compute work integrals around closed loops. Apply Green's theorem to domains with holes. Use it to compute the area.
Compute the divergence and curl of vector fields and the gradient of functions.
Derive simple identities involving div, grad, and curl, for example, product formulas.