Integral over sphere

[Matthiasf]

Hi,

I've been looking the whole day to solve this problem and I cannot find a solution.

I have the general equation of a sphere:

I have also a function that gives a value for every point in the 3D space: f(x,y,z). This function describes amount of light falling in at a point / m².

Now I want to know what is the total amount of light on the sphere. So I have to sum the function over the total area of the sphere. I would think this is an integral of the form int(f(x,y,z) * ds1 *ds2) with ds1 and ds2 some perpendicular vectors in the tangent plane to the sphere in the point x,y,z.

How can I calculate this? I believe I should write ds1 and ds2 in function of x and y and then sum somehow.

Much thanks to the people willing to help me.

Matthias

 

[Steven G]

Well your first problem is that you do not have the general equation of a sphere. Please fix that.

Why would you code your problem by not giving us the definition of f(x,y,z)???

 

[topsquark]

I have the general equation of a sphere:

I have also a function that gives a value for every point in the 3D space: f(x,y,z). This function describes amount of light falling in at a point / m².

To be a bit more direct, your equation is for an ellipsoid, not a sphere. The equation for a sphere is centered on the origin ia [math]x^2 + y^2 + z^2 = r^2[/math]
I agree with Jomo... What is f(x, y, z)?

Since you are only dealing with the surface of the sphere you don't need to worry about r changing, so the differential surface element on the sphere will have dimensions [math](r ~ sin( \theta ) ~ d \theta )*( r ~ d \phi )[/math]. Then integrate over the surface.

-Dan

 

[Matthiasf]

Hi,

Thank you for your replies.

I meant indeed that I have the equation of an ellipsoid. So the equation I gave was correct. The word 'sphere' was not.

I agree with Jomo... What is f(x, y, z)?

f(x,y,z) is a complicated function. It's computed by computer in every point.

Maybe I need to say that I try to solve this by computer. So maybe there is a easier numerical way to do this. So what I was thinking about is to take the value at a point, take this as a constant over a small area and calculate the total amount of light. Then take the next point and so on (see attachment).