Geometrical Significance of determinant of the Jacobian matrix
I know that determinent of the matrix is geometrically thought as the volume of parallelopiped span by the vector.
Is there any geometrical significance of determinant of Jacobian matrix?
Any HElp will be appreciable
Calculus is all about functions which, at very small scales, appear linear. This is why in Calculus I you learn how to find the line tangent to a function by using the derivative at a point. You are finding a linear approximation of your function at a point.
In higher dimensions, a linear map is represented by a matrix, and the Jacobian is that matrix! If you have a differentiable function, no matter how complicated it is, it looks like a linear transformation if you zoom in far enough. If you pick a point $p$, then the Jacobian $J_f(p)$ tells you what that linear transformation is (around $p$). This is analogous to the line approximating your function around a point in calculus I.
What, then, is the determinant of the Jacobian? As you mention, the determinant measures how a linear transformation affects the volume of space. If you look at the cube spanned by the standard basis vectors, we say it has volume 1. After applying our linear transformation $T$, our basis vectors were sent to the edges of a parallelopiped, and the new volume tells you how volumes changed after applying $T$.
In one dimension, if the derivative at a point is $3$, that means it is changing faster than if the derivative were $2$. The determinant of the Jacobian measures the same thing. If $|J_f(p)| = 3$, that means if $C$ is a (very small) cube near $p$, we can expect $f(C)$ to be a parallelopiped with about $3$ times the area of $C$!
This is why, when we change coordinates to take an integral, we need to multiply by a factor of the determinant of the Jacobian of the transformation. We integrate to measure the size of some object, but when we change coordinates the size of that object changes! How does the size change? The determinant of the Jacobian tells us exactly how the size changes at any point.
As an excellent visual example, this is a youtube video showcasing how (at small scales) a differentiable transformation looks linear.
I hope this helps ^_^
