Integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}##

[happyparticle]

Homework Statement

integration

Relevant Equations

F = GMm/r^2

Hi,

I'm trying to integrate $\int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}$ and $\int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}$

I get $\frac{\hat z}{(\sqrt{p^2 + z¸^2)}}$ and $\frac{2z \cdot \hat p}{(\sqrt{p^2 + z¸^2)}}$

But the correct answer should be $\frac{z \hat z}{(\sqrt{p^2 + z¸^2)}}$ and $\frac{2z \cdot \hat p}{(p\sqrt{p^2 + z¸^2)}}$

I'm not sure how to deal with $\hat z$ and $\hat p$

 

[haruspex]

Apart from the unit vectors, your integrals do not seem to have any vector terms. That would make the unit vectors constants which can be moved outside the integral.
It would also mean the two official answers you quote only differ by a constant factor, but the given integrals do not fit that.
Should some of the occurrences of z and p be vectors?

 

[happyparticle]

Sorry, yes z and p are vectors using cylindrical coordinates.

 

[haruspex]

Sorry, yes z and p are vectors using cylindrical coordinates.

But changing all z, p, dz to vectors doesn’t work either.
Is it that $dz\hat z$ is the same as $\vec{dz}$?
If not, please post a version which clarifies it.

 

[happyparticle]

Sorry, I'm not sure to understand. dz is for the derivative of z.

 

[haruspex]

Sorry, I'm not sure to understand. dz is for the derivative of z.

No, it would represent an elemental change in z, but is that regarding z as a vector or as a scalar? An elemental change in $\vec z$ is $\vec{dz}$. If $\vec z$ always points in the same direction then you can write that direction as $\hat z$ and $\vec{dz}$ as $\hat z.dz$, where dz is a scalar.
To make things clear, please write out your integrals showing vectors as appropriate.