Find the divergence of the function

1. Sep 6, 2013

DODGEVIPER13

1. The problem statement, all variables and given/known data
Let V = (sin(theta)cos(phi))/r Determine:
(a) ∇V
(b) ∇ x ∇V
(c) ∇∇V

2. Relevant equations

3. The attempt at a solution

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2. Sep 7, 2013

rude man

Your formulas are all wrong. You need to get the right formulas as you finally did with your gradient exercises.

3. Sep 7, 2013

DODGEVIPER13

I uploaded 2 parts of the answer

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4. Sep 7, 2013

rude man

No time to check it thoroughly now, but 2 cooents:
1. you're using i j k again when you shouldn't.
2. curl(grad f) = 0 always. The curl of the gradient of a function is always zero. (Not all functions have gradients).
3 your writing is hard for me to read but it looks like you wrote grad when you meant div.

5. Sep 7, 2013

DODGEVIPER13

Is my formula even close to correct? If I take off I,j, and k and does it make it a curl when you take the cross product of a gradient?

6. Sep 7, 2013

DODGEVIPER13

Sorry that was one long run on sentence. I meant to say will it be ok if I take off I,j, and k?

7. Sep 7, 2013

rude man

No. The gradient of a function is a vector. Therefore you need the three unit vectors. Do't use i j k unless you're in an xyz cartesian coordinate system. Use the ones I gave you previously.

The curl can only be taken of a vector. There is no such thing as the curl of a scalar.

It's a mathematical identity that curl (grad f) = 0 for any f. It's very good to remember that identity.

8. Sep 7, 2013

DODGEVIPER13

Ok is my formula ok? I'm gonna go look again but I believe Wikipedia had that listed I can add unit vectors that's no problem.

9. Sep 8, 2013

rude man

If wikipedia listed it I'm sure it was right. But they must have had the unit vectors too?
(Now I know how you did so well with the gradients, right?)

10. Sep 8, 2013

DODGEVIPER13

Ugggg that is all Wikipedia lists I'm gonna go to my instructors web page and check there

11. Sep 8, 2013

WannabeNewton

$\hat{i},\hat{j},\hat{k}$ are just labels for unit vectors. They traditionally represent the unit vectors in Cartesian coordinates but you can call them whatever you want, they're just labels. Generally one would write $\hat{e}_1,\hat{e}_2,\hat{e}_3$ to represent unit vectors in arbitrary coordinates. If you are using spherical coordinates, then you need to use the correct formulas for $\nabla f, \nabla \cdot V, \nabla \times V, \nabla^2 f$ in spherical coordinates. You can find them easily online; for example $\nabla = \hat{e}_r\partial_r + \frac{1}{r}\hat{e}_{\theta}\partial_{\theta} + \frac{1}{r\sin\theta} \hat{e}_{\varphi}\partial_{\varphi}$.

12. Sep 8, 2013

rude man

Never mind, I screwed you up & forgot you're now doing divergences, not gradients.

Divergence of a vector is a scalar so you don't attach the unit vectors.

Sorry!

13. Sep 8, 2013

DODGEVIPER13

actually I screwed up the problem never said anything about divergence! That being said I corrected what I did and have uploaded it hopefully its ok!!

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• EPSON018.jpg
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14. Sep 8, 2013

rude man

Oh, OK. I think I mistook this for another of your threads whichI think did have div's in them.

I am too lazy to check all your math steps but for curl[grad(f)] the answer must be zero so I'm afraid something went awry there.

Why don't you run grad(f) thru wolfram alpha to make sure you got that part right.

15. Sep 8, 2013

DODGEVIPER13

ok how do I use wolfram for gradients?

16. Sep 8, 2013

rude man

Stand by, I'll give it a shot, haven't done that myself yet ...

EDIT: OK, punch in " gradient of sin(theta)*cos(phi)/r " in their top window.

They don't include the unit vectors. They give you the r, theta and phi components in that order, separated by commas, instead. Dumb, but better than screwing up solving it ourselves!

Last edited: Sep 8, 2013
17. Sep 8, 2013

DODGEVIPER13

ok uploaded something else I can could use some assistance with

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18. Sep 8, 2013

rude man

Can't read that. What happened with grad f?

Back in 1 hr.

19. Sep 8, 2013

DODGEVIPER13

Evaluate the divergence of the following:
(A) A=xyUx+y^2Uy-xzUz
(B) B=pz^2Up+psin^2(phi)Uphi+2pzsin^2(phi)Uz
(C) C=rUr+rcos^2(theta)Uphi

20. Sep 8, 2013

DODGEVIPER13

The previous post was in reference to my uploaded answer that could not be read.