What are some examples of fields with equal divergences and different curls?

[forty]

Find a pair of fields having equal and divergences in some region, having the same values on the boundary of that region, and yet having different curls.

I really have no idea on where to start for this.

Would making up 2 arbitrary fields in spherical co-ordinates work?

a(theta) + b$$\phi$$ + r$$\hat{r}$$
d(theta) + e$$\phi$$ + f$$\hat{r}$$ (where $$\phi$$ and (theta) are unit vectors, latex isn't working for me >.<)

Then trying to solve for the conditions mentioned?

I know that r$$\phi$$ and r²$$\phi$$ work on the sphere r=1 but I have no idea to go about deriving this. I think this has more to do with me not really grasping vector calculus. Any hints,tips,pointers greatly appreciated.

 

[gabbagabbahey]

Find a pair of fields having equal and divergences in some region, having the same values on the boundary of that region, and yet having different curls.

I really have no idea on where to start for this.

Would making up 2 arbitrary fields in spherical co-ordinates work?

a(theta) + b$$\phi$$ + r$$\hat{r}$$
d(theta) + e$$\phi$$ + f$$\hat{r}$$ (where $$\phi$$ and (theta) are unit vectors, latex isn't working for me >.<)

Then trying to solve for the conditions mentioned?

There are infinitely many solutions and they depend on which region you are interested in, so I would think that "solving" such a system would be difficult.

Instead, you can either start taking educated guesses and then checking that your guess satisfies the given conditions, OR you can appeal to what you know about the physics of electric and magnetic fields...

I know that r$$\phi$$ and r²$$\phi$$ work on the sphere r=1 but I have no idea to go about deriving this.

Okay, this looks like a good "guess". To show that it satisfies the conditions, take the divergence and curl of each field and show that the divergences are equal and the curls are not. Then show that they both have the same value on the boundary.

 

[kuruman]

You know that the divergence of the curl is always zero. If you write

B = A + curlF, then

divB = divA everywhere.

Now curlB must be different from curlA unless curlcurlF=0.
If curlF is zero on the boundary of your region, you're in business.