Why are both r and k in vector notation?

[gtfitzpatrick]

Homework Statement

if \phi = rk/r^{3} where r=xi + yJ + zk and r is the magnitude of r, prove that \nabla\phi = (1/r^{}5)(r^{}2k-3(r.k)r


Right I'm not really sure where to start here...
i know

Magnitude r = \sqrt{x^{2}+y^{2}+z^{2}}

 

[gtfitzpatrick]

\nabla\phi (is generally) = d\phi/dx x + d\phi/dy y + d\phi/dz z

but could someone please give me a pointer from here. can i just multiply across the x, y, z parts of r by k/r^{3} and then differenciate?

 

[BrendanH]

Something I am perhaps missing: why are both r and k in vector notation? This would imply a dot product; do you really mean that?

Also, in your second post, you denote the unit vectors by x,y,z. Do you mean i , j , and k? I'm not trying to be a bug! I just want to try and keep consistent, so that the problem can be clarified.

 

[gtfitzpatrick]

yes in the question they're both in vector notation

 

[BrendanH]

Alright, then that would imply that \phi=z/r^3

 

[gtfitzpatrick]

sorry i meant i, j, k. I'm not sure what you mean...is z is the dot product of r.k?

 

[BrendanH]

Yes, exactly. i dotted with j or k is zero. only k dot k is 1, so the coefficient of z is 1.

 

[gtfitzpatrick]

sorry i don't really follow. I just don't know...

 

[BrendanH]

(xi + yJ + zk) dotted with k = z

After that, it's just a whole of differentiating. Could you more clearly write the 'proof' against which we should be comparing? If need be, dictate what it says instead of trying to write it out.

 

[gtfitzpatrick]

Thanks for you patience with me!
so i now have \phi = z/\sqrt{x^{2}+y^{2}+z^{2}} amd now i just differenciate wrt x then y then z and then tidy it all up?
Thanks a mill for all the help

 

[BrendanH]

yes, except you must cube the <br /> \sqrt{x^{2}+y^{2}+z^{2}}<br /> !

 

[gtfitzpatrick]

Thanks had it, I'm just tidying it up now...you've put me in great humour...thanks a mill...I'm off to see dinosaur jr as soon as i get it tidied up.Thanks again

 

[Marco_84]

or u can use the gradient in spherical coordinates:

http://mathworld.wolfram.com/SphericalCoordinates.html

 

[gtfitzpatrick]

so i differenciated wrt x then y then z and tried to tidy it all up but i got1/r^{5}(-3(r.k)r)

When i differenciated wrt x i got -3x/r^{5} and similar for y and z was this right?

then i just put these answers into \nabla\phi = dx/d\phi i + dy/d\phi j + dz/d\phi k

which gives
-3xz/r^{5} i - 3yz/r^{5} j - 3zz/r^{5} k

am i right so far? it just seemed to tidy up to 1/r^{5}(-3(r.k)r)

when it should be 1/r^{5}(r^{2}k-3(r.k)r)