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

In summary, the conversation discusses the integration of ## \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}## and ## \int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}## in cylindrical coordinates. The given integrals do not have any vector terms except for the unit vectors. After considering the unit vectors as constants that can be moved outside the integral, the two official answers provided differ by a constant factor. However, changing all z and p to vectors does not give the correct answer. It is unclear whether ##dz\hat z## is the same as ##\vec{dz
  • #1
happyparticle
400
20
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##
 
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  • #2
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?
 
  • #3
Sorry, yes z and p are vectors using cylindrical coordinates.
 
  • #4
EpselonZero said:
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.
 
  • #5
Sorry, I'm not sure to understand. dz is for the derivative of z.
 
  • #6
EpselonZero said:
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.
 

1. What does the integral represent?

The integral represents the area under the curve of the given function.

2. How do you solve this integral?

This integral can be solved using the substitution method, where u = p^2 + z^2 and du = 2zdz. After substitution, the integral becomes ∫ 1/u^(3/2) du, which can be easily solved using the power rule.

3. What is the significance of the hat symbol in the integral?

The hat symbol represents a unit vector in the direction of z, which is used to calculate the cross product in the numerator of the integrand.

4. Can this integral be solved using any other method?

Yes, this integral can also be solved using the trigonometric substitution method, where u = tanθ and du = sec^2θ dθ. After substitution, the integral becomes ∫ cosθ/cos^3θ dθ, which can be solved using the power rule and trigonometric identities.

5. Are there any special cases for this integral?

Yes, if p = 0, the integral becomes ∫ zdz/z^3, which is a simple integral that can be solved using the power rule. If z = 0, the integral becomes ∫ 0 dz, which is equal to 0.

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