# Change of coordinates

1. Aug 21, 2015

### Bruno Tolentino

I want to understand the solutions for the PDE of 2nd order $$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} f_x\\ f_y \end{bmatrix} +cf=0$$ But, this depends of the associated conic equation $$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c=0$$ be a parbola, elipse or a hyperbola. One time that it already is known, I think that is necessary to simplify the equation eliminating terms through the change of coordinates.

Derivating the conic wrt x $$\frac{\partial }{\partial x}\left (\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c\right )= \frac{\partial }{\partial x}\left (0 \right )$$ and wrt y $$\frac{\partial }{\partial y}\left (\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c\right )= \frac{\partial }{\partial y}\left (0 \right )$$ obtains other two equations. These two news equations are straight and the intersection between they is the center of the conic of coordinates (β1, β2). So, the system (x, y) is relationed with the new system (x', y') by following vetorial equation $$\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix} \begin{bmatrix} x'\\ y' \end{bmatrix} + \begin{bmatrix} \beta_1\\ \beta_2 \end{bmatrix}$$

Now, is necessary know $$\begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix}$$ How to determinate this matrix? Will be that the Mohr circle can help me?

2. Aug 26, 2015