Calculating area with the metric

[Benjam:n]

So say you have a 2D Riemannian manifold with a metric defined on it and for simplicity let's say its flat. That means there exists a coordinate system for which the metric tensor is the normal Euclidean metric everywhere. However let's say we are using an arbitrary coordinate system with a non Euclidean metric. So we have two vectors whose components are given in this arbitrary coordinate system. To work out the area we must transform both vectors into the coordinate system with the Euclidean metric and then work out the area spanned by the two vectors in that coordinate system, by taking the determinant of the matrix with those two vectors as the columns. My question is, usually in Riemannian geometry we don't know the coordinates with the Euclidean metric in terms of our coordinates so we cleverly remove it from calculations by turning it into the metric which we do have. How do you do that here?

 

[Bill_K]

The invariant volume element in four dimensions is √|g| d⁴x. Likewise for a two-dimensional surface, if you have surface coordinates u¹ and u² and surface metric h_ij, the area element is √|h| du¹ du².