Problem understanding the differential form of the circulation law


by havarija
Tags: circulation, differential, form
havarija
havarija is offline
#1
Jan7-13, 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 .)
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tiny-tim
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#2
Jan7-13, 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 !)
havarija
havarija is offline
#3
Jan7-13, 07:15 AM
P: 2
Now it seems like a ridiculous question :D

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

HallsofIvy
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#4
Jan11-13, 02:38 PM
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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!
joshmccraney
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#5
Jan14-13, 03:24 AM
P: 227
Quote Quote by havarija View Post
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:

[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 u-component of curl.


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