Classical notation for line integrals

In summary, the classical notation for line integrals represents a scalar function as p(x,y)dx + q(x,y)dy, where A = p(x,y) and B = q(x,y) are known as one-forms. This means that if there is a unit displacement in the x-direction, A units of work is done, and if there is a unit displacement in the y-direction, B units of work is done. This interpretation can help clarify the geometrical meaning of the notation.
  • #1
amolv06
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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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  • #2
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.
 
Last edited:
  • #3
Ahh, thanks. This helps me make sense out of what was going on.
 

Related to Classical notation for line integrals

1. What is classical notation for line integrals?

Classical notation for line integrals is a mathematical notation used to represent the integration of a scalar or vector function along a path or curve in a multivariable function.

2. How is classical notation for line integrals written?

Classical notation for line integrals is typically written as ∫C f(x,y) ds, where f(x,y) is the function being integrated, C is the path or curve along which the integral is taken, and ds represents the differential arc length of the curve.

3. What is the significance of the path or curve in classical notation for line integrals?

The path or curve in classical notation for line integrals determines the limits of integration and the direction in which the integration is performed. It also affects the value of the integral, as different paths can yield different results for the same function.

4. How is a line integral evaluated using classical notation?

A line integral using classical notation is evaluated by first parameterizing the path or curve and then substituting the parametric equations into the integral. The integral is then solved using standard integration techniques.

5. What are some real-world applications of classical notation for line integrals?

Classical notation for line integrals is commonly used in physics, engineering, and other scientific fields to calculate quantities such as work, flux, and circulation. It is also used in vector calculus to solve problems involving vector fields and their line integrals.

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