Surface area in spherical co-ordinates

In summary, the conversation discusses the calculation of the fraction of total radiation emitted into a specific region of space using a generalised function in spherical polar coordinates. The solution involves finding the surface area of the function through double integration over elevation and azimuth angles. The speaker is facing difficulties in solving this in spherical coordinates and is seeking help. The formula for the integral is also provided, with the option to replace cosine with sine for a different range of values.
  • #1
FleetFoot
2
0
I have a generalised function in spherical polar co-ordinates that describes the radiant intensity in any given direction of a point source emitter. I need to calculate the fraction of the total radiation that is emitted into a region of space specified by some min to max range of elevation angles phi and a min to max range of azimuth angles theta.

I know that the solution to the problem is essentially finding the surface area of the function by a double integration over phi and theta. However although I know how to solve a generalised surface integral in cartesian co-ordinates I'm not getting anywhere trying to solve this is spherical co-ordinates. Any help would be appreciated.
 
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  • #2
dxdydz=r2drducosvdv, where r is the radial direction, u is "longitude" (full circle), and v is "latitude" (-π/2 < v < π/2). You could replace cosvdv by sinwdw for 0 < w < π.
 

1. What is surface area in spherical coordinates?

Surface area in spherical coordinates refers to the measure of the total area on the surface of a sphere. It is used to calculate the surface area of a three-dimensional object in spherical coordinates, which is a coordinate system that uses angles and distances from a fixed point to describe points in space.

2. How is surface area calculated in spherical coordinates?

The formula for calculating surface area in spherical coordinates is S = 4πr², where S is the surface area and r is the radius of the sphere. This formula can be derived from the surface integral of a sphere in spherical coordinates, which takes into account the angle and distance from the center of the sphere for each point on its surface.

3. What are the advantages of using spherical coordinates for surface area calculations?

Spherical coordinates are advantageous for calculating surface area because they simplify the integration process and allow for a more intuitive understanding of the shape and size of the object. Additionally, they are commonly used in physics and engineering applications, making them a useful tool for solving real-world problems.

4. Can surface area be calculated for irregularly shaped objects in spherical coordinates?

Yes, surface area can be calculated for irregularly shaped objects in spherical coordinates by dividing the object into smaller, more manageable sections and using the formula for surface area of a sphere to calculate the surface area of each section. The surface area of the entire object can then be obtained by summing the surface areas of all the sections.

5. Are there any limitations to using spherical coordinates for surface area calculations?

While spherical coordinates are useful for many applications, they do have limitations, particularly when dealing with very complex or asymmetric shapes. In these cases, other coordinate systems, such as Cartesian coordinates, may be more suitable for calculating surface area. Additionally, spherical coordinates may not be appropriate for objects that are not approximately spherical in shape.

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