Divergence of a Unit Vector

I dont really understand how to do this question. I get that the divergence of the field would be 3, But id have thought the divergence of the unit vector would just be the divergence of the vector itself divided by the magnitude, but it appears that this isnt the case?

Reply 1

Zacken

Original post by Eremor

You can write the unit vector

\hat{v} = \frac{v}{|v|} = \frac{v}{\sqrt{v^2}}

now use the product/quotient rule on this to find the divergence.

If you do it this way - you should end up with

\nabla \cdot \hat{v} = \frac{\nabla \cdot v - 1}{|r|}

(edited 7 years ago)

Reply 2

Math12345

Unit vector W (which is V hat in the question)

= (\frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}})

I will find d/dx for you. The rest is similar.

(Using quotient rule)

\frac{d}{dx}(\frac{x}{\sqrt{x^2+y^2+z^2}}) = \frac{(\sqrt{x^2+y^2+z^2})(1)-(x)(2x*\frac{1}{2}*(x^2+y^2+z^2)^{-\frac{1}{2}})}{x^2+y^2+z^2} = \frac{y^2+z^2}{(x^2+y^2+z^2)^{ \frac{3}{2}}}

You will get

2(x^2+y^2+z^2)

at the top after you find the other derivatives and add them (definition of divergence). Cancel some stuff, you get the answer.

(edited 7 years ago)

Reply 3

atsruser

Original post by Eremor

From a mathematical POV, this is much easier if you convert the field to spherical polars, then find the divergence, since then we get:

\bold{V} = r\bold{\hat{r}}

with no component in the direction of

\bold{\hat{\theta}}, \bold{\hat{\phi}}

Reply 4

M14B

Original post by atsruser

From a mathematical POV, this is much easier if you convert the field to spherical polars, then find the divergence, since then we get:

\bold{V} = r\bold{\hat{r}}

with no component in the direction of

\bold{\hat{\theta}}, \bold{\hat{\phi}}

The divergence in SPC is disgusting

Reply 5

rayquaza17

Original post by M14B

The divergence in SPC is disgusting

Also I think this is an exam question and I imagine they are looking to do it in cartesian since there is information about what divergence in spherical is. Of course OP might have a good memory!!

Reply 6

M14B

Original post by rayquaza17

Also I think this is an exam question and I imagine they are looking to do it in cartesian since there is information about what divergence in spherical is. Of course OP might have a good memory!!

Exactly!

Btw is to possible to sticky again
http://www.thestudentroom.co.uk/showthread.php?t=3330283
as I am finding increasingly difficult to find it down the pages
Thanks
:smile:

Reply 7

rayquaza17

Original post by M14B

Exactly!

Btw is to possible to sticky again
http://www.thestudentroom.co.uk/showthread.php?t=3330283
as I am finding increasingly difficult to find it down the pages
Thanks
:smile:

Done!

Reply 8

M14B

Original post by rayquaza17

Done!

Great

Reply 9

rayquaza17

Original post by M14B

Great

Hey just in case you're looking for it, I've merged the undergrad resources and the a level resources and they're now in a new sticky in maths called "maths resources". :smile:

Reply 10

M14B

Original post by rayquaza17

Hey just in case you're looking for it, I've merged the undergrad resources and the a level resources and they're now in a new sticky in maths called "maths resources". :smile:

No problem

Reply 11

atsruser