Last line integral problem (hopefully)

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Discussion Overview

The discussion revolves around the evaluation of a line integral for the vector field F(x,y) = <2xy-3, x^(2)+4y^(3)+5> and the conditions under which this integral is independent of the path taken. Participants explore concepts related to exact differentials and conservative fields, as well as the necessary conditions for path independence.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests to show that the line integral is independent of path and to evaluate it for specific endpoints.
  • Another participant inquires whether the class has covered the topic of exact differentials.
  • A participant expresses doubt that the concept is known by another name, indicating a potential gap in understanding.
  • One participant offers to provide a detailed explanation of the topic if no one else does, suggesting a willingness to clarify the concept of exact differentials.
  • A participant explains that a line integral is independent of path if there exists a function U such that Fdr is its exact differential, providing a specific function U.
  • Another participant elaborates on the definition of an exact differential and introduces the "cross derivative test" as a method to verify the conditions for path independence.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the understanding of exact differentials, as some express uncertainty about the terminology and concepts involved. The discussion remains unresolved regarding the evaluation of the line integral and the conditions for path independence.

Contextual Notes

Participants have not clarified certain assumptions regarding the definitions of exact differentials and conservative fields. There may be dependencies on the continuity of second partial derivatives that are not fully explored.

PhysicsMajor
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Greetings again,

Show that for F(x,y)=<2xy-3, x^(2)+4y^(3)+5> the line integral F(x,y).dr is independent of path. Then evaluate the line integral for any curve C with initial point (-1,2) and the terminal point (2,3).

Thanks again, you all have been very helpful.
 
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has your class covered exact differentials?
 
i don't believe so unless its known by another name?
 
I'll post a detailed explanation tomorrow (if no one else has by then). I doubt you would know the subject by a different name.
 
Last edited:
A line integral is independent of path if there exists a function U, that Fdr is it's exact (total) differential. In this case U=x^2y-3x+5y+y^4.
dU=(2xy-3)dx+(x^2+4y^3+5)dy=Fdr.
 
An "exact differential" fdx+ gdy (a physics major may prefer to think of it as a "conservative force field") is, as Oggy said, one such that there exist a function U such that dU= fdx+ gdy. By the "chain rule", [tex]dU= \frac{\partial U}{\partial x}dx+ \frac{\partial U}{/partial y}[/tex] so we must have [tex]f= \frac{\partial U}{/partial x}[/tex] and [tex]g= \frac{\partial U}{/partial g}. A quick way to test that is to use the "cross derivative test: If the second partials are continuous, that requires that<br /> [tex]\frac{\partial f}{/partial y}= U_{xy}= U_{yx}= \frac{\partial g}{\partial x}[/itex].[/tex][/tex]
 

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