Problem understanding the differential form of the circulation lawby havarija Tags: circulation, differential, form 

#1
Jan713, 05:50 AM

P: 2

I've encountered a problem in learning about the curl of a vector field.
(My learning material is the "Div, Grad, Curl and all that" from H.M. Shey.) Introduction to problem: The curl of a field F is defined as: ∇×F = i (∂Fz/∂y  ∂Fy/∂z) + j(∂Fx/∂z  ∂Fx/∂x) + k(∂Fy/∂x  ∂Fx/∂y) He claims the following: If we take: n.(∇xF) for n = i, j, k and they all equal 0 that we can conclude that ∇xF = 0 generally. Is it not that we can only conclude that: ∂Fz/∂y = ∂Fy/∂z ∂Fx/∂z = ∂Fx/∂x and ∂Fy/∂x = ∂Fx/∂y Or does this conclusion imply the following somehow? Thanks .) 



#2
Jan713, 06:44 AM

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hi havarija! welcome to pf!
∂Fz/∂y = ∂Fy/∂z ∂Fx/∂z = ∂Fx/∂x and ∂Fy/∂x = ∂Fx/∂y is the same as ∇xF = 0 (and, for any vector V, if n.V = 0 for n = i, j, k then V must be 0 !) 



#3
Jan713, 07:15 AM

P: 2

Now it seems like a ridiculous question :D
Silly me. But now at least I joined the forum :) 



#4
Jan1113, 02:38 PM

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PF Gold
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Problem understanding the differential form of the circulation law
Welcome! I love ridiculous questions I can actually answer some of them!




#5
Jan1413, 03:24 AM

P: 229

[tex]curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr[/tex] where C is a closed loop, A(C) is the area of the loop, vector field F and unit vector u. this is the definition of the ucomponent of curl. 


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