Why is a conserved vector field a gradient of a certain func

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Discussion Overview

The discussion revolves around the concept of conservative vector fields and their relationship to gradients of functions. Participants explore the reasoning behind why a conserved vector field can be expressed as the gradient of a certain function, touching on theoretical aspects and implications in physics.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant notes that a conserved vector field implies the existence of a function whose gradient equals the vector field, seeking to understand the underlying reason for this relationship.
  • Another participant references the work done by conservative forces on closed curves being zero, suggesting this is related to the concept of conservative vector fields.
  • A later reply introduces the "Fundamental Theorem of Calculus" to explain that a conservative force can be viewed as the derivative of an energy function, emphasizing that the integral of a conservative force depends only on the endpoints.

Areas of Agreement / Disagreement

Participants appear to share an understanding of the relationship between conservative vector fields and gradients, but there is no consensus on the deeper reasoning or implications, as questions remain about non-conservative forces.

Contextual Notes

Some assumptions about the definitions of conservative and non-conservative forces are not explicitly stated, and the discussion does not resolve the differences in understanding the implications of these concepts.

Za Kh
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I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.
 
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Hi ZA,

You of course googled conservative vector field but there must be something that isn't clear to you. What specifically ?
 
BvU said:
Hi ZA,

You of course googled conservative vector field but there must be something that isn't clear to you. What specifically ?
Hi! I've figured this out, by the stocks theorem and that the work done by a conservative force on a closed curve is zero but why it isn't true for the non conservative force?
 
That comes from the "Fundamental Theorem of Calculus", that if F is a differentiable function such that F'= f then \int_a^b= f(b)- f(a). A "conservative force" is the derivative of some "energy function". The integral depends only on the end points, not the path between the end points. The integral around a closed path, since the "end points" are the same point, is 0.
 

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