Curl and its relation to line integrals

[member 428835]

hey all

i know and understand the component of curl/line integral relation as: $$curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr$$ where we have vector field $F$, $A(C)$ is the area of a closed boundary, $u$ is an arbitrary unit vector, $dr$ is an infinitely small piece of curve $C$

my question is, how does this definition change if i have, say $curlF\cdot {x}$ versus $curlF\cdot {z}$ where $x$ and $z$ are the unit vectors in the standard cartesian system.

thanks for the feedback! you guys/girls are amazing!

 

[HallsofIvy]

I don't understand your question. First what "definition" are you talking about? The formula you give is not a definition. Second, you are given a formula for $curl F\cdot u$ where u can be any unit vector- but there is no reference to c on the right side- they cannot be equal. Did you mean that u is the unit vector perpendicular to the plane of C? But you did not require that C be a planar curve.

 

[member 428835]

First what "definition" are you talking about? The formula you give is not a definition.

I read information about this on a vector analysis course page by the university of minnesota, where they said the line integral above was the way one formally defines curl. is this incorrect?

Second, you are given a formula for $curl F\cdot u$ where u can be any unit vector- but there is no reference to c on the right side- they cannot be equal. Did you mean that u is the unit vector perpendicular to the plane of C? But you did not require that C be a planar curve.

yes, apologies here. $C$ is a planar closed curve around some point in space orthogonal to $u$

 

[micromass]

I read information about this on a vector analysis course page by the university of minnesota, where they said the line integral above was the way one formally defines curl. is this incorrect?

It's not really incorrect, since it's equivalent to the usual definition (if everything is smooth enough). But it's not the standard definition. Usually textbooks defines curl totally differently. The standard definition is: http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx

I actually do like your limit definition better since it is way more intuitive.

 

[blue_raver22]

The definition of curl and its relation to line integrals does not change if you have different unit vectors such as x and z in the standard cartesian system. The key concept here is that curl is a measure of the rotation or circulation of a vector field, and it is independent of the coordinate system being used. Therefore, the same formula and definition will apply regardless of the unit vectors chosen. The only difference may be in the direction of the resulting vector, but the magnitude and overall concept will remain the same.