Problem understanding the differential form of the circulation law

by havarija
Tags: circulation, differential, form
 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 .)
 Sci Advisor HW Helper Thanks P: 26,167 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 !)
 P: 2 Now it seems like a ridiculous question :D Silly me. But now at least I joined the forum :)
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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!
P: 227
 Quote by havarija 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)
if youre reviewing it may be helpful to consider the following definition of curl component:

$$curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr$$ 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 u-component of curl.

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