Jacobian for a Cartesian to Polar-Coordinate Transformation

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

suppose I have substituted $x=r \cos\theta$ and $y=r \sin\theta$ in an integral to go from cartesian to polar-coordinate. If I use simple area rule or the standard jacobian method, I will get

$dx dy=r dr d\theta$.

At the same time, using the direct method,

$dx=dr \cos\theta-r\sin\theta\; d\theta, \quad $ $dy=dr \sin\theta+r\cos\theta \;d\theta \quad $. Then I find $\bf{dxdy}$ by simply taking the product and neglecting the second order differential, I will get $dxdy=rdrd\theta(cos^{2}\theta-sin^{2}\theta)$. Both results are different. Here there is a missing negative sign and I don't understand it well. This negative sign comes from the evaluation of the determinant, due to its off-diagonal product term of the Jacobian matrix. Hence the right result is $dxdy=rdrd\theta(cos^{2}\theta+sin^{2}\theta)=rdrd\theta$.

I don't understand, why the contradiction comes here ???

The problem is the wrong usage of things like $dx$ and $dy$. People once worked with them as "infinitesimals", but the problem is just that, you can get into confusion pretty quickly. The true rigorous $dx$ and $dy$ are differential forms. They are functions that assign to each point of space one object called alternating tensor. For simplicity, one can consider a tensor to be a multilinear function of vectors, i.e. a function that takes various vectors as parameters, returns numbers and is linear in each parameter with the others held fixed.

The alternating character has to do also with the product of such objects, called the wedge product. This product is such that $dx\wedge dy = -dy\wedge dx$ for example. In your case this is sufficent to establish the fact.

Indeed, the first part of computations is correct:

$$dx = \cos \theta dr-r\sin\theta d\theta,$$

$$dy=\sin\theta dr+r\cos\theta d\theta,$$

now we have

$$dx\wedge dy=(\cos\theta dr-r\sin\theta d\theta)\wedge(\sin\theta dr+r\cos\theta d\theta),$$

but this product is distributive, so that we have

$$dx\wedge dy=(\cos\theta dr)\wedge(\sin\theta dr)+(\cos\theta dr)\wedge(r\cos\theta d\theta)+(-r\sin\theta d\theta)\wedge(\sin\theta dr)+(-r\sin\theta d\theta)\wedge(r\cos\theta d\theta),$$

also scalars can be put outside, so that

$$dx\wedge dy = (\cos\theta\sin\theta)dr\wedge dr+(r\cos^2\theta)(dr\wedge d\theta)-(r\sin^2\theta)d\theta\wedge dr-(r^2\sin\theta\cos\theta)d\theta\wedge d\theta$$

Now, any 1-form $\omega$ satisfies $\omega\wedge \omega = 0$, this is because the alternating property grants that $\omega\wedge\omega=-\omega\wedge\omega$ and so this follows. Because of that, $dr\wedge dr = 0$ and $d\theta\wedge d\theta = 0$. Finally we have

$$dx\wedge dy =r\cos^2\theta dr\wedge d\theta - r\sin^2\theta d\theta\wedge dr,$$

And finally using again the alternating property $-d\theta\wedge dr = dr\wedge d\theta$ and so

$$dx\wedge dy = r\cos^2\theta dr\wedge d\theta + r\sin^2\theta dr\wedge d\theta = r dr\wedge d\theta.$$

Of course, it's not possible to explain everything of differential forms in this single answer, just to show a little of how this fits in your problem. To see more on this, look at Spivak's Calculus on Manifolds (this one is a heavy book), or take a look at "Elementary Differential Geometry" by O'neill, this one has a good introduction to differential forms.

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.