Vector Analysis - Determining whether a vector field is conservative

[Bill Nye Tho]

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.

 

[SammyS]

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 ?

 

[Bill Nye Tho]

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?

 

[Bill Nye Tho]

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.

 

[SammyS]

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.