Wolfram Alpha:

```The Area Metric
---------------

Up until now, the metric has been used in the context of length
associated with a world line.  We now consider the metric in the
context of area associated with a world sheet.  In particular we
want to explore the mapping of a small area on the 2-dimensional
world sheet to an area in n-dimensional spacetime that preserves
the size (i.e. is invariant under the transformation).

Consider the parallelogram PQRT.

R     T
------ ^
/     / |
/θ    /  | h
----    v
P    Q

Area of parallelogram PQRT = PQ*h

= PQ*PRsinθ

= PQ × PR ... the cross product
From the diagram:

Area PQRT = |∂σx × ∂τx|dσdτ

= ∫∫dS

Where dS is the area element of the surface corresponding to
the coordinate increments dσ and dτ.

Area PQRT = ∫∫|∂σx × ∂τx|dσdτ
w

Use the vector identity |a × b| = √(|a|2|b|2 - (a.b)2) to get:

Area PQRT = ∫∫√((∂σx.∂σx)(∂τx.∂τx) - (∂σx.∂τx)2)dσdτ
w

= ∫∫√(g11g22 - g122)dσdτ
w

= ∫∫√(det[g])dσdτ where det[g] = |g11 g12|
w                             |g21 g22|

If g12 = g21 then:

Area PQRT = ∫∫√(det[g])dσdτ
w

= ∫∫dS
w

Where dS = √(det[g])dσdτ

For a 2-D Minkowski spacetime metric det[g] equals -1.
Therefore, it necessary to write:

dS = √(-det[g])dσdτ

Example:

Compute the area of a sphere of radius, r.

φ
|
2π|......
|      :       => Sphere
|   U  :
---------- θ
π

r2 = x2 + y2 + z2

The flat space metric in spherical coordinates with only θ
and φ varying is:

-        -
gmn = | r2   0   |
| 0 r2sin2 |
-        -

Therefore,

det[gmn] = r4sin2θ

Area = ∫∫√(r4sin2θ)dθdφ
U

= 4πr2 after a little math.```