EmmaTassone
  • EmmaTassone
Help with multivariable calculus please.
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
EmmaTassone
  • EmmaTassone
Being F a vector field with continous partial derivatives, If rot(F)=0 and Div(F)=0 is F constant?
EmmaTassone
  • EmmaTassone
I tried using Gauss and stokes theorem but i dont see anything
anonymous
  • anonymous
rot(F) = curl(F)?

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

EmmaTassone
  • EmmaTassone
yep, sry xD here in spanish we call it rot
EmmaTassone
  • EmmaTassone
i think the solution should be find using those theorems
anonymous
  • anonymous
if the curl(F)=0 => F is a conservative vector field
EmmaTassone
  • EmmaTassone
yep, and if Div(F)=0 => F=curl(G) being G another vector field
anonymous
  • anonymous
it's been a while since i've dealt with this stuff so i'm a bit rusty. sorry
IrishBoy123
  • 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
  • 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
  • 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
  • IrishBoy123
@pgpilot326 +1 or its just not actually compressing
IrishBoy123
  • IrishBoy123
and does F is constant mean? like 1,2,3 ?
EmmaTassone
  • EmmaTassone
yep
IrishBoy123
  • IrishBoy123
well we have an example so that is not true
EmmaTassone
  • EmmaTassone
yep thank you very much guys , i really appreciate the help
IrishBoy123
  • IrishBoy123
or think gravity! zero curl, zero divergence.
Michele_Laino
  • 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
  • EmmaTassone
yes and it doesnt have to be necessarily constant, thanks! :)
Michele_Laino
  • Michele_Laino
:)
IrishBoy123
  • 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
  • IrishBoy123
because that nails it :-)
Michele_Laino
  • Michele_Laino
since: \[\Large \nabla \times {\mathbf{F}} = {\mathbf{0}}\]
IrishBoy123
  • 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
  • Michele_Laino
yes! I think so, since the vector equation above, it is an identity
Michele_Laino
  • Michele_Laino
furthermore, also this condition: \[\Large \nabla \cdot {\mathbf{F}} = 0\] holds
IrishBoy123
  • 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
  • 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
  • Michele_Laino
condition*
IrishBoy123
  • 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
  • Michele_Laino
thanks!
IrishBoy123
  • IrishBoy123
you need a medal :-)
IrishBoy123
  • IrishBoy123
or 2
Michele_Laino
  • Michele_Laino
lol!
Michele_Laino
  • Michele_Laino
no worries! I'm happy so! thanks! @IrishBoy123
IrishBoy123
  • IrishBoy123
good!

Looking for something else?

Not the answer you are looking for? Search for more explanations.