# Laplacian and Hessian

1. Jan 5, 2014

### Jhenrique

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?

2. Jan 11, 2014

### tiny-tim

Hello Jhenrique!

The Hessian is a 2x2 matrix, (column-vector)(row-vector).

The Laplacian is a 1x1 matrix, (row-vector)(column-vector).

3. Jan 11, 2014

### dextercioby

The laplacian is the trace of the hessian.