## EmmaTassone one year ago Help with multivariable calculus please.

1. EmmaTassone

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

2. EmmaTassone

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

3. anonymous

rot(F) = curl(F)?

4. EmmaTassone

yep, sry xD here in spanish we call it rot

5. EmmaTassone

i think the solution should be find using those theorems

6. anonymous

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

7. EmmaTassone

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

8. anonymous

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

9. 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

10. 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: $\Large {\mathbf{0}} = {\nabla ^2}{\mathbf{F}}$

11. 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?

12. IrishBoy123

@pgpilot326 +1 or its just not actually compressing

13. IrishBoy123

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

14. EmmaTassone

yep

15. IrishBoy123

well we have an example so that is not true

16. EmmaTassone

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

17. IrishBoy123

or think gravity! zero curl, zero divergence.

18. 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 $$\large x, \; y, \; z$$

19. EmmaTassone

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

20. Michele_Laino

:)

21. 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 $$\nabla \times\vec A = 0$$

22. IrishBoy123

because that nails it :-)

23. Michele_Laino

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

24. 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

25. Michele_Laino

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

26. Michele_Laino

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

27. 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.

28. Michele_Laino

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

29. Michele_Laino

condition*

30. 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.

31. Michele_Laino

thanks!

32. IrishBoy123

you need a medal :-)

33. IrishBoy123

or 2

34. Michele_Laino

lol!

35. Michele_Laino

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

36. IrishBoy123

good!