Surface integrals in spherical coordinates

In summary, the speaker is having trouble calculating flux over sections of spheres without using the divergence theorem. They mention encountering terms like sin^3(phi) and cos^2(phi)sin(theta) that must be integrated and ask if this is normal or if there is an easier method. Another person responds by explaining that integrating sin^3(phi) is not difficult and suggests using u-substitution. The speaker then expresses their frustration with the process.
  • #1
jbusc
211
0
Hi,

I am studying for finals and I'm having trouble calculating flux over sections of spheres. I can do it using the divergence theorem, but I need to know how to do it without divergence thm also.

The problem is, when calculating a vector field such as F(x, y, z) = <z, y, x>, say over the unit sphere (x^2 + y^2 + z^2 = 1), I always end up with weird terms like sin^3(phi) and cos^2(phi)sin(theta) that must be integrated

So, is this normal? Should I memorize integrals for sin^3(phi) and such, or is there an easier method?

thanks
 
Physics news on Phys.org
  • #2
Integrating sin^3(phi) is not difficult. You know sin^3(phi) = sin^2(phi)*sin(phi) = (1-cos^2(phi))sin(phi). Then just use u-substitution with u = 1-cos^2(phi)
 
  • #3
*sigh* I guess it is easy. For some reason it just seems...unnecessary...
 

Related to Surface integrals in spherical coordinates

1. What are surface integrals in spherical coordinates?

Surface integrals in spherical coordinates are mathematical tools used to calculate the flux of a vector field over a surface in three-dimensional space. They involve integrating a given function over a surface, where the surface is defined in terms of spherical coordinates (radius, polar angle, and azimuthal angle).

2. How are surface integrals in spherical coordinates different from other coordinate systems?

In spherical coordinates, the surface to be integrated over is defined by a radius and two angles, whereas in other coordinate systems (such as cartesian or cylindrical), the surface is defined by two parameters (e.g. x and y for cartesian). This means that the surface integrals in spherical coordinates require a different approach and different formulas for calculation.

3. Why are surface integrals in spherical coordinates useful?

Surface integrals in spherical coordinates are useful for calculating the flux of a vector field over a curved surface, which cannot be easily done using other coordinate systems. They are also used in physics and engineering applications, such as calculating the electric field on a spherical surface or the flow of a fluid over a spherical object.

4. How do you set up a surface integral in spherical coordinates?

To set up a surface integral in spherical coordinates, you first need to define your surface in terms of spherical coordinates (i.e. the radius, polar angle, and azimuthal angle). Then, you need to choose an appropriate vector field to integrate over the surface. Finally, you need to use the appropriate formula for surface integrals in spherical coordinates to calculate the flux over the surface.

5. Can surface integrals in spherical coordinates be applied to any type of surface?

Surface integrals in spherical coordinates can be applied to any type of surface, as long as it can be defined in terms of spherical coordinates. This includes curved surfaces, such as spheres, cones, and paraboloids, as well as flat surfaces, such as disks and rectangles. However, the specific formula and approach for calculating the surface integral may vary depending on the type of surface.

Similar threads

Center of mass of spherical shell inside of cone (Apostol Problem).
    • Calculus and Beyond Homework Help
Replies
3
Views
639
Calculate surface integral on sphere
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
How to calculate a sink using spherical coordinates
    • Calculus and Beyond Homework Help
Replies
7
Views
728
Outward flux of a vector field
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Find surface of maximum flux given the vector field's potential
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Surface area of a shifted sphere in spherical coordinates
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Represent a 3d region and compute this triple integral
    • Calculus and Beyond Homework Help
Replies
4
Views
994
Verifying the flux transport theorem
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Surface Integral of a sphere
    • Calculus and Beyond Homework Help
Replies
9
Views
334
Line integral of a scalar function about a quadrant
    • Calculus and Beyond Homework Help
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top