numpy second derivative of a ndimensional array

Question

I have a set of simulation data where I would like to find the lowest slope in n dimensions. The spacing of the data is constant along each dimension, but not all the same (I co

Answer 1

Slopes, Hessians and Laplacians are related, but are 3 different things.
Start with 2d: a function( x, y ) of 2 variables, e.g. a height map of a range of hills,

• slopes aka gradients are direction vectors, a direction and length at each point x y.
  This can be given by 2 numbers dx dy in cartesian coordinates, or an angle θ and length sqrt( dx^2 + dy^2 ) in polar coordinates. Over a whole range of hills, we get a vector field.

• Hessians describe curvature near x y, e.g. a paraboloid or a saddle, with 4 numbers: dxx dxy dyx dyy.

• a Laplacian is 1 number, dxx + dyy, at each point x y. Over a range of hills, we get a scalar field. (Functions or hills with Laplacian = 0 are particularly smooth.)

Slopes are linear fits and Hessians quadratic fits, for tiny steps h near a point xy:

f(xy + h)  ~  f(xy)
        +  slope . h    -- dot product, linear in both slope and h
        +  h' H h / 2   -- quadratic in h

Here xy, slope and h are vectors of 2 numbers, and H is a matrix of 4 numbers dxx dxy dyx dyy.

N-d is similar: slopes are direction vectors of N numbers, Hessians are matrices of N^2 numbers, and Laplacians 1 number, at each point.

(You might find better answers over on math.stackexchange .)