- #1
Jhenrique
- 676
- 4
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}
Gradient:
\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?
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}
Gradient:
\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?