We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...
Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R,
gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}##
,and I just know the metric tensor, but don't know that it is of a sphere.
Now I want to draw a 2D space(surface)...
In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e.
##∂R/(∂z^i )=z_i##
But in curved space (such as, the two dimensional space on a sphere) how can we define covariant basis 'intrinsicly'?(as we have no position...