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

In summary, solving a vector calculus integral over a sphere involves breaking down the integral into smaller components and applying various techniques such as change of variables and integration by parts. The key is to determine the appropriate bounds and use the spherical coordinate system to simplify the integral. By using these techniques and understanding the geometry of the sphere, the integral can be solved efficiently.
  • #1
Hoofbeat
48
0
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!]
 
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  • #2
Isnt n = [tex] \frac{2x,2y,2z}{|gradF|} [/tex]?
 
  • #3
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
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
Theres a few spherical coordinate problems here including derivations that are explained pretty well.

http://tutorial.math.lamar.edu/AllBrowsers/2415/TISphericalCoords.asp
 
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  • #6
thanks. they're still carrying out triple integrals though, whereas I should only be doing a double integral :s
 
  • #7
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.
 
  • #8
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
 
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  • #9
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


~~
 
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Related to How Do You Solve a Vector Calculus Integral Over a Sphere?

1. What is vector calculus?

Vector calculus is a branch of mathematics that studies vector fields and their derivatives, such as gradient, divergence, and curl. It is used to analyze and solve problems in physics and engineering.

2. What are the basic operations in vector calculus?

The basic operations in vector calculus include addition, subtraction, scalar multiplication, dot product, cross product, and differentiation and integration of vector functions.

3. What is the difference between scalar and vector quantities in vector calculus?

Scalar quantities have only magnitude, while vector quantities have both magnitude and direction. In vector calculus, scalar quantities are represented by real numbers, while vector quantities are represented by arrows in space.

4. How is vector calculus used in real-world applications?

Vector calculus is used in a variety of real-world applications such as fluid dynamics, electromagnetism, mechanics, and computer graphics. It is also used in fields such as economics, biology, and finance.

5. What are some common vector calculus theorems and their applications?

Some common vector calculus theorems include the Gradient Theorem, Divergence Theorem, and Stokes' Theorem. These theorems are used to solve problems involving line and surface integrals, flux, and circulation in vector fields.

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