Is there any way to calculate this integral?

[Rafa Ariza]

I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue.
The theory is:

where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2

 

[BvU]

Hello Rafa,

Maybe the problem statement is correct, but you don't tell what you are actually doing (parametric form of $\sigma$ ?)
Maybe the theory is correct, but you don't explain all the symbols, so to me it's of no practical use.
In the past I learned to change to spherical coordinates for something like this and if I do that here I have no problem coming up with the answwer. How about you ?

[edit] Oh, and: for homework you should post in the homework forum and use the template there.

 

[Rafa Ariza]

Hello Rafa,

Maybe the problem statement is correct, but you don't tell what you are actually doing (parametric form of $\sigma$ ?)
Maybe the theory is correct, but you don't explain all the symbols, so to me it's of no practical use.
In the past I learned to change to spherical coordinates for something like this and if I do that here I have no problem coming up with the answwer. How about you ?

[edit] Oh, and: for homework you should post in the homework forum and use the template there.

Thanks,
I want to calculate this integer but σ:x^2+y^2+z^2-R^2=0

 

[ehild]

View attachment 204158
I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue.
The theory is:
View attachment 204159
where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2

You can do the integral in spherical polar coordinates. http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html
Write x and y and also the surface element in the spherical coordinates.

 

[Rafa Ariza]

You can do the integral in spherical polar coordinates. http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html
Write x and y and also the surface element in the spherical coordinates.

Can you write it please?

 

[BvU]

PF requires you to do something too. You said you had done it

I have done it by the parametric form of σ,

Show your work

[edit] on a friendlier note (for a first-time poster): is there something in the link ehild gave that you don't understand ? Do you know what you need from there ?

 

[Rafa Ariza]

PF requires you to do something too. You said you had done it
Show your work

[edit] on a friendlier note (for a first-time poster): is there something in the link ehild gave that you don't understand ? Do you know what you need from there ?

here is my problem but in the other form is resolved yet

 

[BvU]

And what does the theory say about z on $R_{xy}$ ?

 

[Rafa Ariza]

And what does the theory say about z on $R_{xy}$ ?

Rxy is the projection of S in the plane xy.

 

[BvU]

So $R_{xy}$ is a quarter circle (not just the circumference).
But $z=0$ in the plane xy, so what do you do with that $R\over z$ ? Where does this theoretical formula come from ? Is it applicable ?

Would you be interested to follow the path ehild and I learned long ago and proposed here in #2 and #3 ?

 

[Rafa Ariza]

So $R_{xy}$ is a quarter circle (not just the circumference).
But $z=0$ in the plane xy, so what do you do with that $R\over z$ ? Where does this theoretical formula come from ? Is it applicable ?

Would you be interested to follow the path ehild and I learned long ago and proposed here in #2 and #3 ?

yes i am interested. here is the resolution by put the sphere in parametric form r(u,v)

 

[Rafa Ariza]

my problem is to resolve it with surface in this form: G(x,y,z)=x^2+y^2+z^2-R^2=0
the theory is

or equivalent

or

the real problem is in the f(x,y,z(x,y)) or the other equivalent

 

[BvU]

yes i am interested. here is the resolution by put the sphere in parametric form r(u,v)
View attachment 204162

Excellent work.

So -- if the theory is correct and applicable -- we (or rather, your helpers) are back to understanding what is needed for the alternative route.
What is meant with $\left | dG\over dz\right |$ in the formula ?
Does it perhaps mean you need to evaluate this factor in the numerator at $z= \sqrt{R^2 - (x^2+y^2)\,}\$ ?

[edit] oopsed & fixed.

 

[Rafa Ariza]

Excellent work.

So -- if the theory is correct and applicable -- we (or rather, your helpers) are back to understanding what is needed for the alternative route.
What is meant with $\left | dG\over dz\right |$ in the formula ?
Does it perhaps mean you need to evaluate this factor in the numerator at $z= \sqrt{R^2 - (x^2+y^2)\,}\$ ?

[edit] oopsed & fixed.

I will try it, thanks!

 

[Rafa Ariza]

Excellent work.

So -- if the theory is correct and applicable -- we (or rather, your helpers) are back to understanding what is needed for the alternative route.
What is meant with $\left | dG\over dz\right |$ in the formula ?
Does it perhaps mean you need to evaluate this factor in the numerator at $z= \sqrt{R^2 - (x^2+y^2)\,}\$ ?

[edit] oopsed & fixed.

but $x²+y²=R²$ so $z= \sqrt{R^2 - (x^2+y^2)\,}=0$

 

[Rafa Ariza]

this way of solve it is so difficult

 

[BvU]

but $x²+y²=R²$ so $z= \sqrt{R^2 - (x^2+y^2)\,}=0$

No. That is on the rim of the circle, not in the interior. The projection of S on the xy plane is the whole quarter circle, not just the edge.

 

[Rafa Ariza]

No. That is on the rim of the circle, not in the interior. The projection of S on the xy plane is the whole quarter circle, not just the edge.

could you write it ?

 

[BvU]

So your bounds are e.g. 0-1 for x, 0-$\sqrt{1-x^2\,}$ for y

 

[Rafa Ariza]

So your bounds are e.g. 0-1 for x, 0-$\sqrt{1-x^2\,}$ for y

o-R for x right?

 

[BvU]

Yes. You can work R out of the integral and forget about the R⁴ -- it's just a scale factor.

re x, y bounds:
But here too there are more suitable coordinates to be chosen

 

[Rafa Ariza]

Yes. You can work R out of the integral and forget about the R⁴ -- it's just a scale factor.

re x, y bounds:
But here too there are more suitable coordinates to be chosen

so difficult.. i don't know

 

[BvU]

Come on... it looks simple enough now:$$\iint xy\ {R\over \sqrt{R^2-(x^2+y^2)\,}} \ dx dy$$

 

[Rafa Ariza]

Come on... it looks simple enough now:$$\iint xy\ {R\over \sqrt{R^2-(x^2+y^2)\,}} \ dx dy$$

$$R\iint {xy\over \sqrt{R^2-(x^2+y^2)\,}} \ dx dy=\int {r³\over \sqrt {R²-r²\,}} \ dr...$$

 

[BvU]

You can work R out of the integral and forget about the R4 -- it's just a scale factor.

My mistake. #23 is correct, though.

more suitable coordinates

Polar coordinates: $\displaystyle{\int_0^R \int_0^{\pi\over 2}... dr\;d\phi}$

 

[Rafa Ariza]

My mistake. #23 is correct, though.
Polar coordinates: $\displaystyle{\int_0^R \int_0^{\pi\over 2}... dr\;d\phi}$

i edit it

 

[Rafa Ariza]

$\displaystyle{\int_0^R {r³\over \sqrt{R²-r²}\,}\int_0^{\pi\over 2}{\sin \phi\cos \phi}\ dr\;d\phi}$ ?

 

[BvU]

Why the question marks ? The $d\phi$ part we've seen already. To be honest, I have to look up the $dr$ part . How about you ?

 

[Rafa Ariza]

Why the question marks ? The $d\phi$ part we've seen already. To be honest, I have to look up the $dr$ part . How about you ?

YESSSS DONE! THANKS MEN AWESOME

 

[BvU]

You are welcome. Not bad for a first thread !

 

[Rafa Ariza]

You are welcome. Not bad for a first thread !

it was awesome, really love maths