How Do You Solve a Vector Calculus Integral Over a Sphere?

[Hoofbeat]

Anyone take a look at this vector calculus question for me:

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Q. If n is the uni normal to the surface S, evaluate Double Integral r.n dS over the surface of a sphere of radius 'a' centred at the origin.
=====

So I did:

r = (x,y,z)
Sphere: x^2 + y^2 + z^2 = a^2

let f = x^2 + y^2 + z^2

n = gradf / |grad f|

therefore n = (x,y,z)/a

n.r = a

Now how do I proceed with the integral? I thought it would just be

int(2pi->0) int (pi->0) int(a->0) a r.dr.d[theta].d[phi]

which gives the answer [pi]^2.[a]^3 which really doesn't look right! I think it's the actual integral I've made a mistake with! HELP! I HATE VECTOR CALCULUS [and I really need to learn to use latex!]

 

[whozum]

Isnt n = $$\frac{2x,2y,2z}{|gradF|}$$?

 

[Hoofbeat]

Btw, I know that a simple way to solve my problem is just to use the fact that the surface area of the sphere is 4*pi*a^3, but I want to solve the integral explicitly (as practise). Have found my tutor's notes and she found the 'area element' to be a*sin^2*theta*d[theta]*d[phi] and thus carried out a double integral. Could someone explain this please?

 

[Hoofbeat]

Isnt n = $$\frac{2x,2y,2z}{|gradF|}$$?

Yes but the |gradF| on the bottom gives you a factor of 2, so it cancels Having found my tutor's notes I know that I'm doing fine up until the integral and I shouldn't be doing it as a triple integral, but rather a double integral as I've explained in my above post.

 

[whozum]

Theres a few spherical coordinate problems here including derivations that are explained pretty well.

http://tutorial.math.lamar.edu/AllBrowsers/2415/TISphericalCoords.asp

 

[Hoofbeat]

thanks. they're still carrying out triple integrals though, whereas I should only be doing a double integral :s

 

[whozum]

I'd help you out but I don't have my calc 3 book to recall the specific steps in deriving spherical coordinates, but I am pretty sure spherical coordinates only works in triple integrals since it uses 3 parameters, p,theta,phi.

Im probably not the person to helping you with this. Sorry.

 

[whozum]

There is a section on any kind of integration on the left hand navigation menu if you want to look around yourself.

edit: http://tutorial.math.lamar.edu/AllBrowsers/2415/SurfaceIntegrals.asp
Example 2

 

[xanthym]

Anyone take a look at this vector calculus question for me:

=====
Q. If n is the uni normal to the surface S, evaluate Double Integral r.n dS over the surface of a sphere of radius 'a' centred at the origin.
=====

So I did:

r = (x,y,z)
Sphere: x^2 + y^2 + z^2 = a^2

let f = x^2 + y^2 + z^2

n = gradf / |grad f|

therefore n = (x,y,z)/a

n.r = a

Now how do I proceed with the integral? I thought it would just be

int(2pi->0) int (pi->0) int(a->0) a r.dr.d[theta].d[phi]

which gives the answer [pi]^2.[a]^3 which really doesn't look right! I think it's the actual integral I've made a mistake with! HELP! I HATE VECTOR CALCULUS [and I really need to learn to use latex!]

SOLUTION HINTS:
Problem requires evaluation of {∫ ∫ r⋅dA} over Surface of Sphere of Radius "a":
The Unit Area normal element on the Sphere's surface is given by:
dA = r²sin(φ)⋅dθ⋅dφ⋅r/|r|
::: ⇒ ∫ ∫ r⋅dA = ∫ ∫ r⋅r²sin(φ)⋅dθ⋅dφ⋅r/|r| =
= ∫ ∫ r³sin(φ)⋅dθ⋅dφ = ?
The above Double Integral should be evaluated at constant (r = a) for integration limits {0 ≤ θ ≤ 2*π} and {0 ≤ φ ≤ π}.

For more info, see Equation #14 at:
http://mathworld.wolfram.com/SphericalCoordinates.html


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