I will attempt to provide my own perspective. Paralleling @user1620696, I too share the sentiment that differentials should be interpreted not as "infinitesimal geometric quantities", but rather, as functionals, and @user1620696 provided a great description in that differentials are multilinear forms.

I have a simple doubt about the Jacobian and substitutions of the
variables in the integral. @Sijo Joseph

I will only provide a supplement.

From the various advanced undergraduate texts that I have studied, they commonly share the same sentiment that the "$dxdy$" as part of the expression $\iint f(x,y)dxdy$ should not be interpreted, motivationally and initially, on their own, but as interpreted, due to notational conventions, along with the entire expression $\iint f(x,y)dxdy$ on the whole.

Three examples would include Courant's *"Introduction to Calculus and Analysis: Volume II"*, Zorich's *"Mathematical Analysis II"* and Ghorpade's *"A Course in Multivariable Calculus and Analysis"*.

Typically, the authors form a Reimann sum then proceed to demonstrate the existence and uniqueness of the limit of the Reimann sum. Now, proceeding in such a fashion, Courant, for instance, derived the substitution formula with the use of the Jacobian for a system of curvilinear transformation by exploiting the property of the Jacobian that is actually a determinant - a determinant, as most will know, encodes a form of measure that is most commonly interpreted as the volume of an appropriate parallelepiped. The following will make what I have suggested more explicit (from Courant's *"Introduction to Calculus and Analysis II"*, page 399)

$$
\begin{vmatrix}
\phi(u_0+h,v_0) - \phi(u_0, v_0) & \phi(u_0,v_0+k) - \phi(u_0, v_0) \\
\psi(u_0+h,v_0) - \psi(u_0, v_0) & \psi(u_0,v_0+k) - \psi(u_0, v_0) \\
\end{vmatrix}
$$

for $x=\phi(u,v), y=\psi(u,v)$

To interpret the above, consider the point $(u_0,v_0)$; we form a rectangular region with the increments $h$ and $k$, and we perform the corresponding system of curvilinear transformation for the corners of the rectangular region. When we're interested in some form of measure, say, the area of the new region after a curvilinear transformation, the determinant naturally presents itself conveniently, to be used in an *approximation*.

And, by forming a difference quotient (more directly, actually, by applying the intermediate value theorem of differential calculus), we have the following *approximation* (for an appropriate system of curvilinear transformation i.e. e.g. for a system of sufficiently differentiable curvilinear transformation etc.)

$$
hk
\begin{vmatrix}
\phi_u(u_0,v_0) & \phi_v(u_0,v_0)\\
\psi_u(u_0,v_0) & \psi_v(u_0,v_0)\\
\end{vmatrix}
$$

Intuitively, we can treat the increments $h$ and $k$ as $du$ and $dv$, *if the limit exists*, then write $\iint f(x,y)\left | D \right |dudv$ as treated on the whole, and where $D$ is the corresponding Jacobian.

A rigorous proof was, of course, given to support such an intuitive and heuristic consideration.

*Note: It may be noticed that the determinant acts, as a measure, for parallelepipeds. Then why would one so carelessly allow a more general curvilinear transformation? Indeed, the proof in Courant's book took advantage of limiting processes, and taken for granted is the fact that, with appropriate conditions, linear approximations are always possible. Without going into rigorous detail as would bog down this answer, consider Ghorpade's intuitive discussion:*

*The basic idea is to utilize the fact that any "nice" transformation*
*from a subset of $\mathbb{R}^2$ to $\mathbb{R}^2$ can be approximated, at least locally, by an*
*affine transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$.*

*And, in fact, the invariance of areas of parallelepipeds in a 2-dimensional Euclidean space, up to a multiplication by the absolute value of the Jacobian corresponding to the affine transformation, after an affine transformation, was already proven to be true by Ghorparde.*

The Jacobian is quite a fascinating object as it seems to encode a wealth of useful properties. Though, at this stage of my studies, I am limited in scope.