# Proof - Substitution, Jacobian, etc.

• stanley.st
In summary, the conversation discusses a theorem about diffeomorphisms and integrals, and the use of Taylor formula and differential forms to prove it. The speaker also mentions a common mistake in using the wedge product as an inner product.
stanley.st
Hello!

I recently tried to prove following theorem: Let $$\phi:B\to\mathbb{R}^2$$ be a diffeomorphism (regular, injective mapping). Then

$$\int_{\phi(B)}f(\mathbf{x})\,\mathrm{d}x=\int_{B}f(\phi(\mathbf{t}))\left|{\mathrm{det}}\mathbf{J}_{\phi}\right|\mathrm{d}t$$

With following I can't proof this theorem. Look, I start with integral sums

$$(*)\quad\sum_{i=1}^{n}f(x_i,y_j)(x_{i+1}-x_i)(y_{j+1}-y_{j})$$

According to the transformation phi, we have

$$x_i=\phi_x(r(x_i,y_j),t(x_i,y_j))$$$$y_i=\phi_y(r(x_i,y_j),t(x_i,y_j))$$

We can imagine phi as a polar coordinate system transformation, so I use notation with variables r,t. Then we have using Taylor formula

$$\begin{array}{ll}x_{i+1}-x_{i}&=\phi_x(r(x_{i+1},y_j),t(x_{i+1},y_j))-\phi_x(r(x_{i},y_j),t(x_i,y_j))\\&=\frac{\partial \phi_x}{\partial r}(\xi,\eta)(r(x_{i+1},y_j)-r(x_{i},y_j))+\frac{\partial \phi_x}{\partial r}(\xi,\eta)(t(x_{i+1},y_{j})-t(x_{i},y_j))\\&=\frac{\partial \phi_x}{\partial r}\delta r_{i+1,j}+\frac{\partial \phi_x}{\partial r}\delta t_{i+1,j}\quad(\textrm{short form})\end{array}$$

In the same way I can derive

$$y_{j+1}-y_j=\frac{\partial \phi_y}{\partial r}\delta r_{i,j+1}+\frac{\partial \phi_y}{\partial r}\delta t_{i,j+1}$$

If I put this into (*) I get

$$\sum_{i,j=1}^{n}f(\phi(r_{ij},t_{ij}))\left(\frac{\partial \phi_x}{\partial r}\delta r_{i+1,j}+\frac{\partial \phi_x}{\partial r}\delta t_{i+1,j}\right)\left(\frac{\partial \phi_y}{\partial r}\delta r_{i,j+1}+\frac{\partial \phi_y}{\partial r}\delta t_{i,j+1}\right)$$

But this is different than I expected. I expected it in the form like

$$\sum_{i,j=1}^{n}f(\phi(r_{ij},t_{ij}))\left(\frac{\partial \phi_x}{\partial r}\frac{\partial \phi_y}{\partial t}-\frac{\partial \phi_y}{\partial t}\frac{\partial \phi_x}{\partial r}\right)\delta r\delta t$$

Do I something wrong? Thanks a lot..

I don't actually understood what you did there: you confused a lot of coordinate expressions and stated at the end that you expected a formula given by differential forms. I think the best way is to stay within one of the two descriptions, where differential forms are probably easier. We have ##dx \wedge dy## and ask for the transformation to ##d\Phi(x) \wedge d\Phi(y)##. The usual mistake here is to take the wedge product as an inner product, which it is not.

## What is substitution in proof?

Substitution is a technique used in mathematical proofs to replace a variable or expression with another equivalent one. This allows for simplification and manipulation of the original statement.

## What is the Jacobian in proof?

The Jacobian is a matrix of partial derivatives used in multivariable calculus. In proof, it is used to determine the change in variables when performing substitutions and transformations.

## How is substitution used in integration?

In integration, substitution is used to simplify and evaluate complex integrals. By replacing variables with simpler expressions, the integral can be transformed into a more manageable form.

## What is the purpose of the Jacobian in change of variables?

The Jacobian is used in change of variables to account for the change in volume or area when transforming from one coordinate system to another. It is a crucial factor in ensuring the validity of the transformation.

## What is the difference between substitution and change of variables?

Substitution involves replacing variables with equivalent expressions, while change of variables involves transforming the entire coordinate system. Substitution is usually used to simplify and evaluate integrals, while change of variables is used to change the limits of integration and account for the change in volume or area.

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