Miss Millie Millie Line Integrals ~Arc Length~ ≈ !() ! + () ! + () ! ≈ ! ! ! ! + ! ! ! + ! ! ! ∴ = ! ! ! ! ! + ! ! ! + ! ! ! ! ! ~Work Done~ W = F ! ∙ d ! δW = F ! ∙ δr ! ! ! W = ! F ! (x(t), y(t), z(t)) ∙ dr ! ! dt dt !" !" W = ! F ! (x, y, z) ∙ dr ⬚ ! = ! F ! ∙ dr dt dt ⬚ ! remember… r !(t) = x(t)ı̂ + y(t)ȷ̂ + z(t)k ! ∴ dr ! dt = dx dt ı̂ + dy dt ȷ̂ + dz dt k ! ~Path Dependence~ “Work Done’ can be extremely different depending on the path taken. There are some fields, ! , for which ! ! ⋅ ̃ ⬚ ! = 0 for all closed loops. These fields are called conservative, and work done is always 0 ! = − ⏟