Genaral PDE for scalar and vectors

[Jhenrique]

I realized that a PDE of 2nd order can written like: [tex]Af+\vec{b}\cdot\vec{\nabla}f+cf=0[/tex]
$$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}f+\begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}\cdot\begin{bmatrix} \partial_x\\ \partial_y\\ \end{bmatrix}f+cf=0$$
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: $$A\frac{d^2\vec{r}}{dt^2} + B\frac{d\vec{r}}{dt} + C\vec{r}=0$$ and a PDE is: $$a \frac{\partial^2 \vec{\rho}}{\partial u^2} + 2b \frac{\partial^2 \vec{\rho}}{\partial u \partial v} + c\frac{\partial^2 \vec{\rho}}{\partial v^2} + d\frac{\partial \vec{\rho}}{\partial u} + e\frac{\partial \vec{\rho}}{\partial v} + f\vec{\rho}=0$$ 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.

 

[jvicens]

Hello,

Thank you for your question. You are correct in your understanding that a PDE of 2nd order for a vector can be written in the form:

\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}\vec{\rho}+\begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}\cdot\begin{bmatrix} \partial_x\\ \partial_y\\ \end{bmatrix}\vec{\rho}+c\vec{\rho}=0

However, it is important to note that the second-order derivatives in the hessian matrix and the first-order derivatives in the gradient must be applied to each component of the vector separately. In other words, the hessian and gradient operators are not applied to the entire vector at once, but rather to each component individually.

To make this more clear, let's consider an example. Let's say we have a vector function \vec{\rho}(x,y) = (x^2,y^2). The hessian matrix for this vector function would be:

\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}\vec{\rho} = \begin{bmatrix} 2 & 0\\ 0 & 2\\ \end{bmatrix}

Similarly, the gradient of this vector function would be:

\begin{bmatrix} \partial_x\\ \partial_y\\ \end{bmatrix}\vec{\rho} = \begin{bmatrix} 2x\\ 2y\\ \end{bmatrix}

So, if we were to apply this PDE to our vector function, it would look like:

\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} 2 & 0\\ 0 & 2\\ \end{bmatrix}\begin{bmatrix} x^2\\ y^2\\ \end{bmatrix}+\begin{bmatrix} b_1\\ b_2\\ \end{bmatrix}\cdot