Dear Lazyweb,
Most of you know that dy/dx means limΔx→0 (y(x+Δx)−y(x))/Δx . Similarly most of you know ∂y/∂x means the same thing (at least in one variable). But what does dy mean on its own? We use it all the time in an integral, but is there some limit for which it can be substituted? What about ∂y?
Extra credit for differential geometry folks: Suppose V is a solid 3-D volume. I often see ∂V representing its 2-D perimeter - the surface of V. How rigorous is this notation? (i.e., is it a natural extension of the ∂ operator onto 3-D sets or just a notatnional convenience?) Compare and contrast the meaning of ∂V with that of dV with respect to the meaning of the ∂ and d operators.
Extra credit for differential geometry folks: Suppose V is a solid 3-D volume. I often see ∂V representing its 2-D perimeter - the surface of V. How rigorous is this notation? (i.e., is it a natural extension of the ∂ operator onto 3-D sets or just a notatnional convenience?) Compare and contrast the meaning of ∂V with that of dV with respect to the meaning of the ∂ and d operators.