Vector Analysis - Determining whether a vector field is conservative

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Homework Help Overview

The discussion revolves around determining whether a vector field is conservative, focusing on the conditions involving the curl of the vector field and the equality of partial derivatives. Participants explore the implications of these conditions in the context of vector analysis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question whether calculating the curl of a vector field is the most reliable method for determining conservativeness. They also consider if checking the equality of partial derivatives is equally valid and discuss potential limitations of these methods.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the conditions for a vector field to be conservative. Some guidance has been offered regarding the implications of the partial derivatives and the curl, but no consensus has been reached.

Contextual Notes

Participants note that the absence of a sketch of the vector field may affect their analysis. There is also mention of the specific dimensionality of the vector fields being considered, particularly in relation to the XY, XZ, and YZ planes.

Bill Nye Tho
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Homework Statement



n/a

Homework Equations



∇ x F = 0

∂Q/∂x = ∂P/∂y

The Attempt at a Solution



n/a

Given that no sketch of the vector field is given;

Is determining the curl of a vector field the most fail proof of determining whether it is conservative?

I'm just wondering whether or not determining ∂Q/∂x = ∂P/∂y is just as fail proof (Given that: F=Pi + Qj + Rk) because it seems like a faster method within the boundary of this course.
 
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Bill Nye Tho said:

Homework Statement



n/a

Homework Equations



∇ x F = 0

∂Q/∂x = ∂P/∂y

The Attempt at a Solution



n/a

Given that no sketch of the vector field is given;

Is determining the curl of a vector field the most fail proof of determining whether it is conservative?

I'm just wondering whether or not determining ∂Q/∂x = ∂P/∂y is just as fail proof (Given that: F=Pi + Qj + Rk) because it seems like a faster method within the boundary of this course.
What if ∂Q/∂x = ∂P/∂y, but ∂Q/∂z ≠ ∂R/∂y ?
 
SammyS said:
What if ∂Q/∂x = ∂P/∂y, but ∂Q/∂z ≠ ∂R/∂y ?

Then the partials of Q and P will only be effective with i + j vector fields?
 
Also, the answer to your question would be that the field would only be conservative in the XY plane but not in the XZ or YZ.
 
Bill Nye Tho said:
Also, the answer to your question would be that the field would only be conservative in the XY plane but not in the XZ or YZ.
I've not aware of that sort of distinction.

If ∂Q/∂z ≠ ∂R/∂y, then ∇ x F ≠ 0 , so the field, F is not conservative.
 

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