## curl and its relation to line integrals

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!
 Recognitions: Gold Member Science Advisor Staff Emeritus 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.

 Quote by HallsofIvy 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?

 Quote by HallsofIvy 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 $u$

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