Finding Nabla Operator for f(r) with r = |R|

[MCooltA]

Homework Statement

I need to find \nablaf(r). I am given r = |R| where R is a vector, R =(x,y,z). I also have the function f(r) which is a differentiable function of r.

Homework Equations

So i know \nabla(g) = (\partialg/\partialx, \partialg/\partialy, \partialg/\partialz)

The Attempt at a Solution

So I've got;

r=√(x^2 + y^2 + z^2)

\nabla(r) = (2x,2y,2z)

Do i need to apply \nabla operator again to the above \nabla(r)?

Does \nabla(\nabla(r)) equal \nablaf(r)?

 

[HallsofIvy]

No, of course not. For one thing, \nabla \nabla r makes no sense. \nabla r is a vector function and "\nabla f has to be applied to a scalar function. Also, there is NO "f" in that. You would be saying that nabla f(r) is the same for all functions! You need the chain rule:
\nabla f(r)= \frac{df}{dr}\nabla r
df/dr will be a scalar function aand \nabla f a vector function so this is scalar multiplication.

Or, equivalently, replace r in the function with (x^2+ y^2+ z^2)^{1/2} and find the gradient.

What is f?

 

[MCooltA]

Thanks for your reply. So far I've got;

\nabla(r) = (1/r) (x,y,z).

And then for df/dr = \partialf_{x}/\partialx + \partialf_{y}/\partialy + \partialf_{z}/\partialz

Im unsure about what df/dr is.