Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplacian and Hessian

  1. Jan 5, 2014 #1

    I was studying matrix calculus and learned new things as:
    [tex]\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}[/tex]
    [tex]\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}[/tex]
    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. jcsd
  3. Jan 11, 2014 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hello Jhenrique! :smile:

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

    The Laplacian is a 1x1 matrix, (row-vector)(column-vector). :wink:
  4. Jan 11, 2014 #3


    User Avatar
    Science Advisor
    Homework Helper

    The laplacian is the trace of the hessian.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook