Classical notation for line integrals

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SUMMARY

The discussion focuses on the classical notation for line integrals, specifically the representation of a scalar function f(x,y) as p(x,y)dx + q(x,y)dy. Participants clarify that this notation corresponds to a one-form, where A = p(x,y) and B = q(x,y). The geometrical interpretation involves understanding that unit displacements in the x and y directions correspond to work done, represented by A and B, respectively. This explanation aids in comprehending the underlying meaning of the notation.

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  • Understanding of scalar functions in two dimensions
  • Familiarity with differential forms and one-forms
  • Basic knowledge of line integrals in calculus
  • Concept of work done in physics
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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.
 
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Its because [tex]A \ dx + B \ dy[/tex] is known as a one-form. In this case [tex]A = p(x,y)[/tex] aand [tex]B = q(x,y)[/tex]. 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.
 
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Ahh, thanks. This helps me make sense out of what was going on.
 

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