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$$^{}2$$k-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.

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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 $$\sqrt{x^{2}+y^{2}+z^{2}}$$ !

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

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)