Calculating Area and Volume of a sphere through line element

In summary, the problem involves calculating the area and volume of a line element in flat space-time with polar coordinates. The line element is described as ds2= -dt2+dr2+r2(dθ2+sin2θdΦ2), and the equations dA = √g11g22 dx1 dx2 and dV = √g11g22g33 dx1 dx2dx3 are provided as tools for the calculation. However, the attempt at a solution provided does not match the actual answers, and there seems to be confusion about which variables correspond to g1, g2, and g3. It is suggested to use sqrt(g2 g3) d theta d phi to calculate dA
  • #1
Tony Stark
51
2

Homework Statement


Flat space-time in polar coordinate is considered. The line element is
ds2= -dt2+dr2+r2(dθ2+sin2θdΦ2)

The actual answers are given below, but I can't come up to them. Need urgent help.

Homework Equations


dA = √g11g22 dx1 dx2
dV = √g11g22g33 dx1 dx2dx3

The Attempt at a Solution


g1= 1
g2= r2
g3= r2sin2θ

⇒dA = √1.r2 dr.dθ
dA= r dr dθ
(ACTUAL ANSWER= dA = r2 sinθ dθ dΦ)

Cant calculate Volume
(ACTUAL ANSWER= dV= r2sinθ dθ dΦ dr)
 
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  • #2
The area is surface area, not area of the projection. That is why the answer is in ##d\theta d\phi##.
What variables do your g1, g2, g3 correspond to? It seems like g1 is the dr term, g2 is the d\theta term, and g3 is the d\phi term.
Why can't you calculate the volume?
 
  • #3
RUber said:
The area is surface area, not area of the projection. That is why the answer is in ##d\theta d\phi##.
What variables do your g1, g2, g3 correspond to? It seems like g1 is the dr term, g2 is the d\theta term, and g3 is the d\phi term.
Why can't you calculate the volume?
Why would the answer be dΘdΦ instead of drdΦ?
G1,G2,G3 are basis four vector, then how could they be dr,dθ and dΦ?
Explanation needed.
 
  • #4
Perhaps I am unfamiliar with your application. However, these look like spherical coordinates to me, and the surface area does not require a change in r.
g1 is 1, which corresponds to the square of the factor used for a change in r dr. g2 is r^2, which corresponds to the square of the factor used for a change in theta d theta.
g3 is r^2 sin^2 theta which corresponds to the square of the factor used for a change in phi d phi.
So, from what you have shown, it seems clear that you should be using sqrt(g2 g3) d theta d phi to calculate dA.
 
  • #5
What is the problem statement you were given, word for word? You seem to be omitting important details.
 
  • #6
vela said:
What is the problem statement you were given, word for word? You seem to be omitting important details.
According to the question, I had to calculate the area and volume of line element described above.
 
  • #7
May I have certain more specification into the mistake I am doing...:bow:
 

FAQ: Calculating Area and Volume of a sphere through line element

1. How do you calculate the area of a sphere through line element?

To calculate the area of a sphere through line element, you can use the formula A = 4πr², where A is the area and r is the radius of the sphere. This formula is derived by integrating the line element of a sphere, which is equal to 2πr.

2. What is the line element of a sphere?

The line element of a sphere is a differential element that represents the length of a small portion of the sphere's circumference. It is equal to 2πr, where r is the radius of the sphere.

3. How do you find the volume of a sphere through line element?

The volume of a sphere can be calculated by integrating the line element of the sphere, which is equal to 4πr². This gives us the formula V = (4/3)πr³, where V is the volume and r is the radius of the sphere.

4. Can you calculate the area and volume of a sphere with a non-uniform radius?

Yes, the formulas for calculating the area and volume of a sphere through line element can be applied to spheres with non-uniform radii. However, you will need to use calculus to integrate the varying line element in order to get an accurate result.

5. What are the units used to measure area and volume of a sphere through line element?

The units used to measure area of a sphere through line element are square units, such as square meters (m²) or square inches (in²). The units used to measure volume are cubic units, such as cubic meters (m³) or cubic inches (in³).

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