How Do You Calculate the Normal Vector of a Sphere in Spherical Coordinates?

In summary, the conversation discusses converting a given sphere into spherical coordinates and finding the normal of a surface using the gradient. The method for finding the normal in polar coordinates is also mentioned.
  • #1
brunette15
58
0
I am given the sphere V= x^2 + y^2 + z^2 =< 1

I have converted it to spherical coordinates:

x = rsin(t)cos(f)
y = rsin(t)cos(f)
z = rcos(t)

where t ranges from 0 to pi, and f ranges from 0 to 2pi.

I am unsure how to go about this problem from here. Any guidance would be really appreciated :)
 
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  • #2
brunette15 said:
I am given the sphere V= x^2 + y^2 + z^2 =< 1

I have converted it to spherical coordinates:

x = rsin(t)cos(f)
y = rsin(t)cos(f)
z = rcos(t)

where t ranges from 0 to pi, and f ranges from 0 to 2pi.

I am unsure how to go about this problem from here. Any guidance would be really appreciated :)

Hey brunette15! ;)

We can find the normal of a surface $f(x,y,z)=0$ by taking the gradient $\nabla f$... (Thinking)
 
  • #3
I like Serena said:
Hey brunette15! ;)

We can find the normal of a surface $f(x,y,z)=0$ by taking the gradient $\nabla f$... (Thinking)

Hi again I like Serena :)

I am aware that we could do that but for this particular case i am trying to prove the divergence theorem of Gauss :/
 
  • #4
brunette15 said:
Hi again I like Serena :)

I am aware that we could do that but for this particular case i am trying to prove the divergence theorem of Gauss :/

Huh? :confused:

What do you want to do then?

Do you want to find the normal in polar coordinates?
If so, the method is the same - we just need the gradient in polar coordinates instead of cartesian coordinates.
 

Related to How Do You Calculate the Normal Vector of a Sphere in Spherical Coordinates?

1. What is the normal vector of a sphere?

The normal vector of a sphere is a vector that is perpendicular to the tangent plane at any given point on the surface of the sphere. It points directly away from the center of the sphere.

2. How is the normal vector of a sphere calculated?

The normal vector of a sphere can be calculated by taking the gradient of the equation that represents the sphere, which is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 where (a,b,c) is the center of the sphere and r is the radius. The result will be a vector with components (x-a, y-b, z-c) that is normalized to have a magnitude of 1.

3. Can the normal vector of a sphere change at different points on the surface?

No, the normal vector of a sphere is constant at every point on its surface. This is because the sphere is a symmetrical shape and the gradient of its equation remains the same at all points.

4. What is the significance of the normal vector of a sphere?

The normal vector of a sphere is important in many applications, such as computer graphics and physics. It is used to determine the direction of light reflection and refraction, as well as the direction of force on a particle moving on the surface of the sphere.

5. Can the normal vector of a sphere be negative?

Yes, the normal vector of a sphere can have negative components. This simply means that the vector is pointing in the opposite direction, away from the center of the sphere. However, the magnitude of the vector will always be positive.

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