Shape of volume element in curvilinear system? I have always pictured volume element as a small cuboid in with volume $dx dy dz$. however in curvilinear system, how would the shape of this volume element be?
I mean in spherical polar coordinate system, how the shape of this volume element be visualized (is it a small sphere whose integration give the volume of object or same), or my idea of 3D Cartesian coordinate absolutely wrong.
 A: the easiest way of picturing the volume element is to start in a point $x\in M$ given in local coordinates $x^\mu$ and go along a (maybe curved) line $x^\mu + dx^\mu$.
So, lets do this! In a two dimensional polar coordinate system you can start at some point that is given by a radius $r = r_0$ and some angle $\varphi = \varphi_0$. Now, you draw a small arc segment from $(r_0, \varphi_0)$ to $(r_0, \varphi_0 + d\varphi)$ where $d\varphi$ is just a small angle and you hold the radius constant.
Then, you do the same from  $(r_0, \varphi_0)$ to $(r_0 + dr, \varphi_0)$. Now you mark the point $(r_0 + dr, \varphi_0 + d\varphi)$ and connect this point (in the same manner as you drew the segments before) to the endpoints of your lines.
Now, can you calculate the surface area of this little element?
If you have done this, can you relate your calculation to the volume form $\mathrm{vol} = \sqrt{g} dy^1\wedge\dots\wedge dy^n = r dr \wedge d\varphi$?
Sincerely
Robert
PS.: I left out the orientation of the manifold for clearity.
