- #1
Jhenrique
- 676
- 4
Hellow!
I was studying matrix calculus and learned new things as:
\frac{d\vec{y}}{d\vec{x}}=\begin{bmatrix} \frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\ \frac{dy_2}{dx_1} & \frac{dy_2}{dx_2} \\ \end{bmatrix}
\frac{d}{d\vec{r}}\frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2} = \begin{bmatrix} \frac{d^2}{dxdx} & \frac{d^2}{dydx}\\ \frac{d^2}{dxdy} & \frac{d^2}{dydy}\\ \end{bmatrix}
Those are the real definition for Jacobian and Hessian. However, the definition for Laplacian is ##\triangledown \cdot \triangledown = \triangledown^2##, that corresponds to ##\frac{d}{d\vec{r}} \cdot \frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2}##, but this definition conflicts with the definition for Hessian that is ##\frac{d^2}{d\vec{r}^2}## too. So, where is the mistake with respect to these definitions? I learned something wrong?
I was studying matrix calculus and learned new things as:
\frac{d\vec{y}}{d\vec{x}}=\begin{bmatrix} \frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\ \frac{dy_2}{dx_1} & \frac{dy_2}{dx_2} \\ \end{bmatrix}
\frac{d}{d\vec{r}}\frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2} = \begin{bmatrix} \frac{d^2}{dxdx} & \frac{d^2}{dydx}\\ \frac{d^2}{dxdy} & \frac{d^2}{dydy}\\ \end{bmatrix}
Those are the real definition for Jacobian and Hessian. However, the definition for Laplacian is ##\triangledown \cdot \triangledown = \triangledown^2##, that corresponds to ##\frac{d}{d\vec{r}} \cdot \frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2}##, but this definition conflicts with the definition for Hessian that is ##\frac{d^2}{d\vec{r}^2}## too. So, where is the mistake with respect to these definitions? I learned something wrong?
