Solving the Problem: Deriving ∂2φ/∂x2

[whatisreality]

Homework Statement

Let φ=φ(r) and r=√x²+y²+z². Find ∂²φ/∂x².
Show that it can be written as (1/r + x²/r³)∂φ/∂r + x²/r²∂²φ/∂r².

Homework Equations

Use the identity ∂r/∂x = x/r.

The Attempt at a Solution

I think I know ∂φ/∂x. Using the chain rule, it's ∂r/∂x ∂φ/∂r. That gives x/r ∂φ/∂r. If that's wrong it might be because I know you have to take account of all dependences, but I don't actually know how to.
So assuming that's ok, I then need to use the product rule, and that gave me
1/r ∂φ/∂r + ∂φ/∂x ∂φ/∂r ∂r/∂x.
Which I know is wrong, because it's a show that question!

 

[Mark44]

Homework Statement

Let φ=φ(r) and r=√x²+y²+z². Find ∂²φ/∂x².
Show that it can be written as (1/r + x²/r³)∂φ/∂r + x²/r²∂²φ/∂r².

Homework Equations

Use the identity ∂r/∂x = x/r.

The Attempt at a Solution

I think I know ∂φ/∂x. Using the chain rule, it's ∂r/∂x ∂φ/∂r. That gives x/r ∂φ/∂r.

I would write this as dφ/dr ∂r/∂x or φ'(r)∂r/∂x = dφ/dr (x/r). φ is a function of r alone, so the derivative for this function is the regular derivative instead of the partial derivative. Now use the product rule to get ∂²φ/∂x².

If that's wrong it might be because I know you have to take account of all dependences, but I don't actually know how to.
So assuming that's ok, I then need to use the product rule, and that gave me
1/r ∂φ/∂r + ∂φ/∂x ∂φ/∂r ∂r/∂x.
Which I know is wrong, because it's a show that question!

 

[whatisreality]

I would write this as dφ/dr ∂r/∂x or φ'(r)∂r/∂x = dφ/dr (x/r). φ is a function of r alone, so the derivative for this function is the regular derivative instead of the partial derivative. Now use the product rule to get ∂²φ/∂x².

I'm doing something wrong. So I got ∂²φ/∂x² = 1/r ∂φ/∂r + ∂/dr (dφ/dr) ∂r/∂x x/r. This doesn't give me the answer, it gives
∂²φ/∂x² = 1/r dφ/dr + x²/r² ∂²φ/∂r², but I really can't see what's wrong!

 

[whatisreality]

I'm missing a whole term: x²/r³ dφ/dr.

 

[Mark44]

I'm doing something wrong. So I got ∂²φ/∂x² = 1/r ∂φ/∂r + ∂/dr (dφ/dr) ∂r/∂x x/r. This doesn't give me the answer, it gives
∂²φ/∂x² = 1/r dφ/dr + x²/r² ∂²φ/∂r², but I really can't see what's wrong!

You don't show the work leading up to this, but I'm guessing that you did this: $\frac{\partial}{\partial x} \frac{x}{r} = \frac{1}{r}$. If so, that's wrong, since you would be treating r as a constant. In fact, r is a function of x (and y and z).

 

[whatisreality]

Ohhh... That's exactly what I did. I always forget that!

 

[whatisreality]

Should be y²+z²/r³ then.
Yep, that gives the right answer! Thank you, I was getting really confused.