The last time I did any vector calculus was many years ago. I've just revisited it and been looking at line integrals over vector fields, e.g. $$\int_{\partial D} \mathbf{F}\cdot\text{d}\mathbf{r}=\int\int_D\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\text{d}t,$$ where $\mathbf{F}$ is a vector field, $\mathbf{r}$ is the chosen parametrization and $\partial D$ is the boundary of the region $D$. I know vector calculus goes on to define the $\text{curl}$ and $\text{div}$ operators and derive some useful results about these. So far I've only encountered vectors, but I was wondering does advanced study of vector calculus consider the possibility of $\mathbf{F}$ being a matrix? Does anyone know of a good resource or references?
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There is a Wikipedia Article called "Matrix Calculus". https://en.wikipedia.org/wiki/Matrix_calculus – Fly by Night Jan 10 '14 at 23:30
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Your formula is a particular case of Stoke's theorem inter dimensions 1 and 2, and since the matrices are also vectors, for a matrix $2\times 2$, surely, there are some versions in dimension 3 and 4 or 4 and 5. – janmarqz Jan 10 '14 at 23:30
