Gradient theorem, why F=-grad(U) ?

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Discussion Overview

The discussion revolves around the relationship between the mathematical expression of the gradient theorem and its physical interpretation, specifically why forces are represented as F = grad(U) in mathematics and F = -grad(U) in physics. The scope includes theoretical considerations and conceptual clarifications related to conservative forces and potential energy.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the difference in sign between the mathematical and physical representations of force, questioning if it relates to Newton's third law.
  • Another participant notes that only conservative forces can be expressed as gradients of scalar fields, contrasting them with non-conservative forces like friction.
  • A participant explains that in physics, the negative sign accounts for changes in potential energy rather than kinetic energy, while in mathematics, the sign is not flipped.
  • One participant emphasizes that forces are directed downhill in a physical context, while the mathematical perspective may consider the gradient as pointing uphill.

Areas of Agreement / Disagreement

The discussion reveals differing perspectives on the interpretation of the gradient theorem in mathematics versus physics. Participants do not reach a consensus on the implications of the sign difference, indicating ongoing debate.

Contextual Notes

The discussion highlights the potential for confusion arising from the different contexts in which the gradient theorem is applied, particularly regarding the definitions of conservative forces and the implications of potential energy changes.

amiras
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I am having difficulties to understand why in mathematics when calculating line integrals using gradient theorem we use F=grad(U), and in physics it is always F=-grad(U)? It seems important to me, because I may end up getting answer with opposite sign.

Is it somehow related to Newton's third law?
 
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Only those forces we call "conservative" will be gradients of scalar fields.
You'll meet other types of forces that are not conservative, for example forces of friction.
 
If we put them into a common context, then F is a force field. A line integral of F represents the work done by the force field on a free particle that traverses that path. The amount of work done is equal to the particle's gain in kinetic energy, which is equal to its loss of potential energy. So when physicists flip the sign, they are accounting for changes in potential energy rather than changes in kinetic energy.

Outside this physical context, there is no reason to flip the sign. You just use the fundamental theorem of line integrals.
 
Vargo said:
If we put them into a common context, then F is a force field. A line integral of F represents the work done by the force field on a free particle that traverses that path. The amount of work done is equal to the particle's gain in kinetic energy, which is equal to its loss of potential energy. So when physicists flip the sign, they are accounting for changes in potential energy rather than changes in kinetic energy.

Outside this physical context, there is no reason to flip the sign. You just use the fundamental theorem of line integrals.

Thank you! This is exactly what I wanted!
 
Because forces point downhill. In math, the purer idea is uphill.
 

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