Gradients vs. Partial Derivatives

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Discussion Overview

The discussion revolves around the distinction between partial derivatives and gradients, particularly in the context of deriving a function related to air convection involving temperature and pressure as variables.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant asks about the difference between partial derivatives and gradients in the context of deriving a function "f(T,P)" for air convection.
  • Another participant states that a gradient is the matrix containing all the partial derivatives.
  • A further contribution explains the gradient for a function of three variables and describes the relationship between the differential and the gradient.
  • A repeated inquiry emphasizes the difference, suggesting that partial derivatives are "limits" while the gradient is characterized as an operator.

Areas of Agreement / Disagreement

No consensus is reached on the definitions or implications of partial derivatives and gradients, as multiple perspectives are presented without resolution.

Contextual Notes

Participants express varying interpretations of the concepts, indicating potential limitations in definitions and the mathematical context in which these terms are applied.

shanepitts
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What is the difference between partial derivatives and gradients, if there is any?

I'm asking because I need to derive a function " f (T,P) " for air convection; where T is temperature and P is pressure and both are variables in this case.

Thanks
 
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A gradient is the matrix containing all the partial derivatives.
 
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For a function of three variables, grad F(x,y,z)= \nabla F(x, y, z)= \frac{\partial F}{\partial x}\vec{i}+ \frac{\partial F}{\partial y}dy\vec{j}+ \frac{\partial F}{\partial z}\vec{j}. In particular, the differential, dF= \frac{\partial F}{\partial x}dx+ \frac{\partial F}{\partial y}dy+ \frac{\partial F}{\partial z}dz, can be thought of as the dot product of \nabla F and dx\vec{i}+ dy\vec{j}+ dz\vec{k}.
 
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shanepitts said:
What is the difference between partial derivatives and gradients, if there is any?

I'm asking because I need to derive a function " f (T,P) " for air convection; where T is temperature and P is pressure and both are variables in this case.

Thanks
partial derivatives are "limits" meanwhile the gradient is an operator.
 

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