Express derivatives most intuitively

1. Feb 28, 2013

Gavroy

I have given a function g(t)=∇(f(x(t))) , f: IR³->IR and x: IR-> IR³ and want to express the first 3 derivatives with respect to time most simply.
I thought that g'(t)=Hessian(f(x(t)))dx/dt
but how do I get the further derivatives. is there any chance to express those in terms of taking the gradient or the hessian matrix of a function several times?

2. Feb 28, 2013

HallsofIvy

Staff Emeritus
If f is a function from R3 to R3, then its derivative is, as you say, a linear transformation from R3 to R3 which you can represent as a 3 by 3 matrix, the Hessian, at each point in R3, which has 9 entries and so can be thought of as in R9.

And that, in turn, means that the derivative function is from R3 to R9 so its derivative would be a linear tranformation from R3 to R9 which could be represented by a "3 by 3 by 3" matrix- a sort of three dimensional variation of a matrix. Higher derivatives, then, would involve higher dimensions.

3. Feb 28, 2013

Gavroy

okay, thank you for your reply, but how do I write down these higher derivatives explicitely, so that I get the right answer?