Help with multivariable calculus please.

- EmmaTassone

Help with multivariable calculus please.

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- EmmaTassone

Being F a vector field with continous partial derivatives, If rot(F)=0 and Div(F)=0 is F constant?

- EmmaTassone

I tried using Gauss and stokes theorem but i dont see anything

- anonymous

rot(F) = curl(F)?

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## More answers

- EmmaTassone

yep, sry xD here in spanish we call it rot

- EmmaTassone

i think the solution should be find using those theorems

- anonymous

if the curl(F)=0 => F is a conservative vector field

- EmmaTassone

yep, and if Div(F)=0 => F=curl(G) being G another vector field

- anonymous

it's been a while since i've dealt with this stuff so i'm a bit rusty. sorry

- IrishBoy123

1) curl F = 0 is not the definitive test for a conservative field: http://mathinsight.org/path_dependent_zero_curl
2) i've had a look about for examples and \((2x,2y,−4z)\) has zero curl and divergence

- Michele_Laino

hint:
if we apply this vector identity:
\[\Large \nabla \times \left( {\nabla \times {\mathbf{F}}} \right) = \left( {\nabla \cdot {\mathbf{F}}} \right)\nabla - {\nabla ^2}{\mathbf{F}}\]
using your condition, we have:
\[\Large {\mathbf{0}} = {\nabla ^2}{\mathbf{F}}\]

- anonymous

curl is tendancy to rotate and divergence is compressibility. if they're both 0, doesn't that just mean that an incompressible fluid is flowing in a way that it tends not to rotate?

- IrishBoy123

@pgpilot326 +1
or its just not actually compressing

- IrishBoy123

and does F is constant mean?
like 1,2,3 ?

- EmmaTassone

yep

- IrishBoy123

well we have an example so that is not true

- EmmaTassone

yep thank you very much guys , i really appreciate the help

- IrishBoy123

or think gravity!
zero curl, zero divergence.

- Michele_Laino

we can show, that, from the condition:
\[\Large {\mathbf{0}} = {\nabla ^2}{\mathbf{F}}\]
the general field \( \large {\mathbf{F}}\) depends linearly on the cartesian coordinates \( \large x, \; y, \; z \)

- EmmaTassone

yes and it doesnt have to be necessarily constant, thanks! :)

- Michele_Laino

:)

- IrishBoy123

@Michele_Laino
interesting
why did you set the triple product to zero ?
and what does \(\nabla \times ( \nabla \times\vec A)\) actually mean if we already know that \( \nabla \times\vec A = 0\)

- IrishBoy123

because that nails it :-)

- Michele_Laino

since:
\[\Large \nabla \times {\mathbf{F}} = {\mathbf{0}}\]

- IrishBoy123

indeed. the curl is zero so can we use the bac cab rule for anything?
like 2 x 0 = 3 x 0
2 = 3

- Michele_Laino

yes! I think so, since the vector equation above, it is an identity

- Michele_Laino

furthermore, also this condition:
\[\Large \nabla \cdot {\mathbf{F}} = 0\]
holds

- IrishBoy123

aaahhh! i think i am getting you.
so, ok, ....the gravitational field, inverse square but no curl no divergence....
i hate to think how you fit that into cartesian to get a linear.

- Michele_Laino

I have developed, component by component the cndition:
\[\Large {\mathbf{0}} = {\nabla ^2}{\mathbf{F}}\]
using this other condition:
\[\Large \nabla \times {\mathbf{F}} = {\mathbf{0}}\]

- Michele_Laino

condition*

- IrishBoy123

yep, i get that bit
the triple product is zero because AxB is zero and .....
if div F is zero, the laplacian must also be zero
so we have a linear in x,y,z
i get the reasoning, sure. it's very good.

- Michele_Laino

thanks!

- IrishBoy123

you need a medal :-)

- IrishBoy123

or 2

- Michele_Laino

lol!

- Michele_Laino

no worries! I'm happy so! thanks! @IrishBoy123

- IrishBoy123

good!

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