Classical notation for line integrals

[amolv06]

I don't really understand the classical notation for line integrals, namely why would you want to represent a scalar function f(x,y) as p(x,y)dx + q(x,y)dy. I also don't fully understand the geometrical interpretation of this. Though solving the problems is easy, I don't really understand what it means. The notation f(x,y)ds seems far more intuitive to me. Can anyone link me to a geometrical interpretation for the classical notation of line integrals? Thanks.

 

[tronter]

Its because A \ dx + B \ dy is known as a one-form. In this case A = p(x,y) aand B = q(x,y). Think of it like this: If there is unit displacement in the x-direction then A units of work is done. If there is a unit displacement in the y-direction then B units of work is done.

 

[amolv06]

Ahh, thanks. This helps me make sense out of what was going on.