Vector Calculus Question

  • Thread starter Hoofbeat
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  • #1
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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!]
 

Answers and Replies

  • #2
2,210
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Isnt n = [tex] \frac{2x,2y,2z}{|gradF|} [/tex]?
 
  • #3
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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?
 
  • #4
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whozum said:
Isnt n = [tex] \frac{2x,2y,2z}{|gradF|} [/tex]?


Yes but the |gradF| on the bottom gives you a factor of 2, so it cancels :frown: 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.
 
  • #5
2,210
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Theres a few spherical coordinate problems here including derivations that are explained pretty well.

http://tutorial.math.lamar.edu/AllBrowsers/2415/TISphericalCoords.asp [Broken]
 
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  • #6
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thanks. they're still carrying out triple integrals though, whereas I should only be doing a double integral :s
 
  • #7
2,210
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I'd help you out but I dont have my calc 3 book to recall the specific steps in deriving spherical coordinates, but im 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.
 
  • #8
2,210
1
There is a section on any kind of integration on the left hand navigation menu if you wanna look around yourself.

edit: http://tutorial.math.lamar.edu/AllBrowsers/2415/SurfaceIntegrals.asp [Broken]
Example 2
 
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  • #9
xanthym
Science Advisor
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Hoofbeat said:
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


~~
 
Last edited:

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