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Genaral PDE for scalar and vectors

  1. Apr 1, 2014 #1
    I realized that a PDE of 2nd order can written like: [tex]A:Hf+\vec{b}\cdot\vec{\nabla}f+cf=0[/tex]
    [tex]\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}f+\begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}\cdot\begin{bmatrix} \partial_x\\ \partial_y\\ \end{bmatrix}f+cf=0[/tex]
    But and if f is a vector, how would be a general PDE of 2nd order for a vector?

    EDIT: Let me be more clear with my ask:

    An ODE of 2nd order for a vector is: [tex]A\frac{d^2\vec{r}}{dt^2} + B\frac{d\vec{r}}{dt} + C\vec{r}=0[/tex] and a PDE is: [tex]a \frac{\partial^2 \vec{\rho}}{\partial u^2} + 2b \frac{\partial^2 \vec{\rho}}{\partial u \partial v} + c\frac{\partial^2 \vec{\rho}}{\partial v^2} + d\frac{\partial \vec{\rho}}{\partial u} + e\frac{\partial \vec{\rho}}{\partial v} + f\vec{\rho}=0[/tex] But how write this PDE in the correct way? Because if I replace the scalar ##f## (in the 1st equation above) by vector position ##\vec{\rho}##, I'll have a direct product between the hessian ##H## and ##\vec{\rho}##, and another between ##\vec{\nabla}## and ##\vec{\rho}##, thing that I never saw.
     
    Last edited: Apr 1, 2014
  2. jcsd
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