Potential Energy in 3D: Partial Derivatives

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Discussion Overview

The discussion revolves around the use of partial derivatives versus total derivatives in the context of finding potential energy in three dimensions. Participants explore the implications of these mathematical approaches within the framework of potential fields and vector fields, particularly in gravitational contexts.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions why partial derivatives are preferred over total derivatives when determining potential energy in three dimensions.
  • Another participant seeks clarification on which specific formula is being referenced in the discussion.
  • A third participant asks for clarification on what derivatives are being discussed.
  • A later reply suggests that taking partial derivatives allows for the derivation of a vector field from a potential field, specifically relating to gravitational acceleration and potential per unit mass.
  • This reply also notes that if the vector field is conservative, one can assign a zero potential at a point and use a path integral to find the potential field.
  • The same participant raises a question about the meaning of taking a total derivative of a function of three variables, suggesting that it would require selecting a path or tangent, whereas partial derivatives align with coordinate axes.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants express differing levels of understanding and seek clarification on the concepts involved. Multiple viewpoints regarding the use of derivatives remain present.

Contextual Notes

Participants do not provide specific definitions or assumptions regarding the terms used, which may lead to ambiguity in the discussion. The implications of conservative fields and the nature of potential energy are not fully resolved.

Gurasees
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To find potential energy in 3 dimensions why do we take partial derivative and not total derivative?
 
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Which formula do you mean?
 
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Derivatives of what?
 
Gurasees said:
To find potential energy in 3 dimensions why do we take partial derivative and not total derivative?
Guessing at the context...

If you have the potential field (e.g. gravitational potential per unit mass as a function of position), you can take three partial derivatives to arrive at the associated vector field (e.g. gravitational acceleration as a function of position).

If you have the vector field (e.g. gravitational acceleration as a function of position) and an assurance that the field is conservative then you can arbitrarily assign a zero potential somewhere, and take a path integral along an arbitrarily chosen path to obtain the associated potential field (e.g. gravitational potential per unit mass as a function of position).

What would it mean to take the total derivative of a function of three variables? The only way I see to do it would be to choose a path (or at least a tangent to a path). Taking a partial derivative amounts to picking a tangent that is aligned with a chosen coordinate axis.
 

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