Calculating Area and Volume of a sphere through line element

[Tony Stark]

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)

 

[RUber]

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?

 

[Tony Stark]

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.

 

[RUber]

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.

 

[vela]

What is the problem statement you were given, word for word? You seem to be omitting important details.

 

[Tony Stark]

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.

 

[Tony Stark]

May I have certain more specification into the mistake I am doing...