## Problem understanding the differential form of the circulation law

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 .)
 PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in $403M deal (Update)>> LA to give every student an iPad;$30M order>> CIA faulted for choosing Amazon over IBM on cloud contract
 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor 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 !)
 Now it seems like a ridiculous question :D Silly me. But now at least I joined the forum :)

Recognitions:
Gold Member
$$curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr$$ 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.