Integrating spherical normal components

In summary, the normal vector at a point on a hemisphere can be expressed in terms of the unit vectors in the x, y, and z directions. The normal vector can be integrated over the surface of the hemisphere to find the surface area.
  • #1
mewmew
114
0
I am trying to integrate the the normal vector to the surface of a hemisphere but am having some trouble. I am pretty sure that the x and y components will cancel out, and that the z component in spherical coordinates is R*Cos[theta] but for some reason I am having trouble really understanding how to do it.

I know this is really basic but I'm horrible at vector calculus. Any help is greatly appreciated. Thanks
 
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  • #2
Why would the x and y components cancel out? Is this a special 3-D situation?
 
  • #3
Well, can you express the normal vector at (R,theta,phi) in terms of the unit vectors in the x, y, and z directions? Then, you can break up the integral into the sum of 3 integrals, and since the unit vectors are constant, you can factor them out of the integrals.
 
  • #4
berkeman said:
Why would the x and y components cancel out? Is this a special 3-D situation?
Yes, it's the upper unit hemisphere! For every vector xi+ yj+ zk there will be a vector -xi- yj+ zk.

mewmew, by "the normal vector" do you mean the unit normal vector? And are you integrating each component of the vector independently or are you integrating [itex]\vec{v}\cdot d\vec{S}[/itex]?
 
  • #5
Sorry, my terminology isn't that great. Ultimately I am trying to find < n_z >, which I believe is n_z integrated over my surface, divided by the area. Also "n_z" is the unit normal vector in the z direction, I tried to do it in tex but was having problems.

Well, can you express the normal vector at (R,theta,phi) in terms of the unit vectors in the x, y, and z directions? Then, you can break up the integral into the sum of 3 integrals, and since the unit vectors are constant, you can factor them out of the integrals.
Since the unit vectors come out of the integrals then the integrals should just give me the surface area, correct? So <n_z> would just be the unite vector in the z direction? I am probably way off basis but my intuition tells me that would be the case...although it is wrong quite a lot!
 
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  • #6
No, n_z should be the portion of the unit normal in the z direction, right? The unit normal at a specified point on the surface can be written as n = n_x i + n_y j + n_z k, where n_x^2 + n_y^2 + n_z^2 = 1 (it's a unit normal!).

Suppose we were just dealing with a circle in polar coordinates. At any point on the circle (r, theta), you can write the unit normal as n = i cos theta + j sin theta, where i is the unit vector in the x direction and j is the unit vector in the y direction. On your hemisphere, you should be able to write a similar formula the normal in terms of i, j, k, theta, and phi. Then, you toss *this* vector into a double integral with appropriate limits (and R^2 sin theta dtheta dphi). Since this is a sum, you can break it up into three integrals, factor out the unit vectors and find (as you expected) the integrals for i and j vanish and you are left with your value for k.

Make sense?
 
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Related to Integrating spherical normal components

What is the definition of "integrating spherical normal components"?

Integrating spherical normal components refers to the process of combining the individual components of a spherical normal vector (such as x, y, and z coordinates) into a single vector quantity.

Why is integrating spherical normal components important in scientific research?

Integrating spherical normal components is important because it allows for a more comprehensive understanding of a vector's direction and magnitude, which is crucial in many scientific fields such as physics and engineering.

How is integrating spherical normal components different from integrating other types of vector components?

The main difference is that spherical normal components are based on a spherical coordinate system, while other types of vector components may be based on Cartesian or other coordinate systems. This means that the equations and methods used for integration may differ.

What are some common applications of integrating spherical normal components?

Integrating spherical normal components is commonly used in fields such as computer graphics, geophysics, and astronomy. It is also essential in navigation and GPS systems, as well as in the analysis of fluid flow and electromagnetic fields.

What are some challenges that may arise when integrating spherical normal components?

One challenge is ensuring that the coordinate systems used for the individual components are consistent and properly aligned. Another challenge is dealing with singularities, which can occur when the radius of the sphere approaches zero. Additionally, integration may become more complicated when dealing with non-uniform spherical grids.

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