Simple question about cross product properties

[Dvsdvs]

Hello, this is pretty straightforward. I need to take cross product:
kq(x/(x^2+y^2),y/(x^2+y^2)) x (d/dx,d/dy)

since kq is a scalar can i just leave it outside the calculation until the very end and for now just calculate the cross product of the two and once i get a definite answer, multiply that answer by kq. Basically can i do this:

((x/(x^2+y^2),y/(x^2+y^2)) x (d/dx,d/dy)) dot kq

 

[gabbagabbahey]

Yes, that's fine. (However, writing "dot" kq is confusing since the dot product is undefined between a scalar and a vector...I would write it as $kq\left((x/(x^2+y^2),y/(x^2+y^2))\times (d/dx,d/dy)\right)$ instead)

 

[Dvsdvs]

yeah thank you. would you know if it would help to convert to polar coordinates when calculating the flux?

 

[Mark44]

Am I missing something here? It looks like you are crossing two vectors in R². The cross product I know about is defined only for two vectors in R³.

 

[gabbagabbahey]

yeah thank you. would you know if it would help to convert to polar coordinates when calculating the flux?

It depends on which surface you are trying to calculate the flux through.

Also, are you actually trying to calculate the curl of (x/(x^2+y^2),y/(x^2+y^2))? If so, you need to realize that $\vec{nabla}\time\vec{F}$ is very different from $\vec{F}\times\vec{\nabla}$! The former is the curl of F, which will be a vector. While the latter is a vector differential operator which operates on some vector.

 

[Dvsdvs]

well surface is x^2+y^2=a^2. and for curl ill add a 0 vector for z and the del operator will be (d/dx, d/dy, d/dz) making it R3. and i actually have to prove its conservative away from the region. so i will take $$\nabla$$ X F in that order yeah i realize i have it swapped before hand. I don't really know what to do about the flux though. Thank you so much for help by the way

Edit-im gettng something weird for the k vector of cross product= (y/(x^2+y^2)) d/dx - (x/(x^2+y^2)) d/dy how does this equal 0? NVM, i see it after quotient rule

 

[Dvsdvs]

alrite so i said flux=$$\int$$$$\int$$$$\nabla$$dotF dA.
since i just proved that dot product = 0. I am left with

$$\int$$$$\int$$ (0) dx dy over the domain: x^2 + Y^2=a^2
sooo, what now

 

[gabbagabbahey]

$$\int_{\mathcal{S}}\vec{\nabla}\cdot\vec{F}dA$$ Does not represent the flux of F through $\mathcal{S}$...It doesn't even represent the flux of div(F) since the flux of a scalar is not well defined (to the best of my knowledge anyways).

I think perhaps you should post the entire original problem word for word so that we can see exactly what you are supposed to compute...

 

[Dvsdvs]

The 2D vector F = kq((x)/(x^2+y^2), (y)/(x^2+y^2))

Show that this vector eld is both conservative, and divergence free
away from the origin (0; 0). Done dot product=0 and cross product=0

Find the
Find the flux of this vector field over the circle C given by x2+y2 = a2 also says that flux is integral of F.Nds but in my book it says that this is equal to
double integral of the dot product of del operator FdA

and yeah youre right that integral doesn't but the double integral with the Domain being C of that does represent the flux

 

[Dvsdvs]

and this Use the Divergence theorem to show that the flux will be the same
even if the curve C is not a circle, but a reasonably nice curve enclosing the origin
(closed, orientable, simple (i.e. doesn't cross itself), simply connected, enclosing the
origin). But this only makes things worse so...

 

[gabbagabbahey]

...but in my book it says that this is equal to
double integral of the dot product of del operator FdA

No, I'm sure your book doesn't say that. What your book probably says is:

$$\oint_{\mathcal{S}}\vec{F}\cdot\hat{N}dS=\int_{\mathcal{V}}(\vec{\nabla}\cdot\vec{F})dV$$

Where $\mathcal{S}$ is the closed surface that bounds the volume $\mathcal{V}$

right? (this is known as the divergence theorem, which is sometimes called "Gauss' theorem" or "Green's theorem")

But(!) this doesn't help you here since the surface $x^2+y^2\leq a^2$ is just a disk of radius $a$; which is an open surface.

Instead you have to calculate the flux $$\int_{\mathcal{S}}\vec{F}\cdot\hat{N}dS$$ directly.

Begin by expressing the unit normal to the surface in Cartesian unit vectors...does F have any component parallel to the normal n this case? If not, then F.NdS=0 and so the flux is zero.

 

[Dvsdvs]

oooo alright i see this now thanks!

 

[gabbagabbahey]

and thus Use the Divergence theorem to show that the flux will be the same
even if the curve C is not a circle, but a reasonably nice curve enclosing the origin
(closed, orientable, simple (i.e. doesn't cross itself), simply connected, enclosing the
origin). But this only makes things worse so...

OH!

You should not have left this part of the problem description out of your previous post!

After reading this, it seems that they want you to use the 2D version of the divergence theorem. The version I posted above is the 3D version.

And the "flux" they want you to calculate is also the 2D version of flux; so instead of calculating $$\int_{\mathcal{S}} \vec{F}\cdot\hat{k}dxdy$$ over the disk, they actually want you to calculate $$\int_{\mathcal{C}} \vec{F}\cdot\hat{r}rd\theta$$ over the circle (I'm using $r$ and $\theta$ to represent polar coordinates, and $\hat{r}$ to represent the radial unit vector)

And that is what you are supposed to apply the divergence theorem to.

The point of the problem is to show that div(F) is not actually zero everywhere, and is in fact infinite at the origin.

 

[Dvsdvs]

but i can't calculate the flux at the origin. as the vector is undefined...so how do u take dot product of this $$\nabla$$dot F when F doesn't even exist at the origin

 

[gabbagabbahey]

but i can't calculate the flux at the origin. as the vector is undefined...so how do u take dot product of this $$\nabla$$dot F when F doesn't even exist at the origin

You can't calculate $\vec{\nabla}\cdot\vec{F}$ directly (at the origin). But you can calculate

$$\int_{\mathcal{C}} \vec{F}\cdot\hat{N}ds$$

You will get the same non-zero answer no matter what curve $\mathcal{C}$ you use (as long as it encloses the origin exactly once); and from that you can conclude that $\vec{\nabla}\cdot\vec{F}$ cannot be zero at the origin.

Then, if you have been taught about the dirac delta function, you should be able to figure out what $\vec{\nabla}\cdot\vec{F}$ actually is.