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It is not clear to me why it is incorrect to substitute for ##dx## and ##dy##. Could you please explain elaborately?

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...