So I saw a video by 3b1b on how derivative operators can be expressed by a square matrix. I applied this concept to vector valued functions in polar coordinates. This is what I have so far:
$$ \vec{v}=v_r\hat{r}+v_\theta\hat{\theta}=\begin{bmatrix}v_r\\v_\theta\end{bmatrix} $$ $$ \vec{r}=|\vec{r}|\hat{r} $$ $$ \hat{r}=cos\theta+sin\theta $$ $$ \hat{\theta}=-sin\theta+cos\theta $$ $$ \frac{d}{dt}=\begin{bmatrix}\frac{d■}{dt} & -\frac{d\theta}{dt}■\\\frac{d\theta}{dt}■ & \frac{d■}{dt}\end{bmatrix} $$ $r$ is the position vector.
I give up on making that look good but anyways. The black squares are placeholders for when you operate this on anything. to operate anything it is just multiplication with this but instead of multiplying the cells you compose them instead. here's some examples: $$ \frac{d■}{dt}f(t)=\frac{df}{dt}\qquad \frac{d■}{dt}\frac{d■}{dt}=\frac{d^2■}{dt^2} $$
To make it so that the matrix derivative operator doesn't lose it's properties when operating on other matrices all matrix cells must have a placeholder.
My question is is what kind of object is this? I know it's a linear operator already but I don't know too much more of it's properties. I'm pretty sure the time integral can be represented as a matrix but I have yet to construct it.