Problem understanding the differential form of the circulation law

[havarija]

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 .)

 

[tiny-tim]

welcome to pf!

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 !)

 

[havarija]

Now it seems like a ridiculous question :D

Silly me. But now at least I joined the forum :)

 

[HallsofIvy]

Welcome! I love ridiculous questions- I can actually answer some of them!

 

[member 428835]

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.