# Rotational in terms of vector calculus

1. Jan 12, 2014

### Jhenrique

Hellow!

I was noting that several definitions are, in actually, expressions of vector calculus, for example:

Jacobian:
$$\frac{d\vec{f}}{d\vec{r}}=\begin{bmatrix} \frac{df_1}{dx} & \frac{df_1}{dy} \\ \frac{df_2}{dx} & \frac{df_2}{dy} \\ \end{bmatrix}$$
Hessian:
$$\frac{d^2f}{d\vec{r}^2} = \begin{bmatrix} \frac{d^2f}{dxdx} & \frac{d^2f}{dydx}\\ \frac{d^2f}{dxdy} & \frac{d^2f}{dydy}\\ \end{bmatrix}$$
$$\frac{df}{d\vec{r}}=\begin{bmatrix} \frac{df}{dx} & \frac{df}{dy} \end{bmatrix}$$
Divergence:
$$trace\left ( \frac{d\vec{f}}{d\vec{r}} \right ) = trace \left ( \begin{bmatrix} \frac{df_1}{dx} & \frac{df_1}{dy} \\ \frac{df_2}{dx} & \frac{df_2}{dy} \\ \end{bmatrix} \right )$$
Laplacian:
$$trace\left ( \frac{d^2f}{d\vec{r}^2} \right ) = trace \left ( \begin{bmatrix} \frac{d^2f}{dxdx} & \frac{d^2f}{dydx}\\ \frac{d^2f}{dxdy} & \frac{d^2f}{dydy}\\ \end{bmatrix} \right )$$

However, still remained a doubt, is possible to express the rotational in terms of vector/matrix calculus, like above?

2. Jan 14, 2014

### Jhenrique

OMG! I forgot that in english you do not speak "rotational" and yes "curl". My question is wrt curl ...