1 Calculate the divergence and curl of the vector field F(x,y,z) = 2xi+3yj+4zk.
2 Find the potential of the function for the conservative vector field:
F(x,y,z) = (y+z)i + (x+z)j + (x+y)k.
3 Use Green's theorem to calculate the work done when a particle is moved along the helix x = cos t, y = sin t, z = 2t from (1,0,0) to (1,0,4pi) against the force
F(x,y,z) = -yi +xj +zk.
4 Calculate the outward flux of the vector field F = xi + yj +zk across the closed surface S, the boundry of the solid paraboloid bounded by the xy - plane and z = 9 - x^2 - y^2.
The solution covers several basic properties of vector fields: divergence, curl, flux, potential function. It also explains what a conservative vector field. The detailed and step-by-step explanations provide the students a clear prospective of properties of the vector field.