Understanding the Curl Theorem: Examples and Explanation

[kliang1234]

Hi, this is a very simple question about the curl theorem. It says in my book:

" If F is a vector field defined on all R3 whose component functions have continuous partial derivatives and curl F = 0 , then F is a conservative vector field"

I might sound stupid, but what exactly does "defined on all R3" and "continuous partial derivatives" mean? Can you also provide examples?

I tried to apply it to a problem:

If Curl <0, z/(z^2 + y^2) , -y/(z^2+y^2) > = 0 , is the vector field conservative?

The answer is no. Can anyone explain why its not conservative.
It is continuously differentiable, and curl F = 0. To me, it appears to satisfy the criteria of the theorem.

Please help, i have a midterm tomorrow.

 

[dextercioby]

Defined on all R^3 means that it is defined on all points of R^3, in particular it is defined in the origin of a coordinate system on R^3, namely in the point (0,0,0) and moreover, it must have continuous partial derivatives at any point of R^3, particularly in (0,0,0).

Question: Is your vector defined all over R^3 ?

 

[kliang1234]

I see. The vector is not R3 because its not defined at the origin because of the j and k components. Correct?

 

[dextercioby]

It's not defined in the origin, indeed. Its domain is simply $R^3 -\{(0,0,0)\}$, so it doesn't fulfil the definition you had there.

 

[kliang1234]

dextercioby, you are god.
thank you so much