# Homework Help: Notation for third order derivative of a vector function

1. Jun 19, 2010

### patiobarbecue

1. let f: R^n -> R, then f' is a vector and f'' is a matrix, how about f'''? it is a cube? I guess we have to use matrix notation for f'''. I have seen the notation " f'''(x)(h,h,h) ", which is a real number for sure. I have no clue how to operate it though. Any reference on third order derivative and its notation?

2. Relevant equations

3. f'' is a matrix, then f''' looks like a cube, where the third dimension are vectors of the following elements: d^3 f/dx_i dx_j dx_1, d^3 f/dx_i dx_j dx_2, ..., d^3 f/dx_i dx_j dx_n
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 19, 2010

### Hurkyl

Staff Emeritus
Actually, f'(x) is not a vector -- it's a linear form. Specifically, it's the linear form with the property that
$$f'(x)(v) = \nabla_v f(x)$$​
Of course, once you've chosen an inner product, you can convert linear forms into vectors and vice versa, and forget there's a distinction.

e.g. if vectors are Nx1 "column vectors", then f'(x) is a row vector. With respect to the dot product, we can transpose f'(x) to get a vector.

Similarly, f''(x) is not a matrix -- it's a bilinear form.
$$f''(x)(v,w) = \nabla_v \nabla_w f(x)$$​
But you can convert bilinear forms into matrices and vice versa.

f'''(x) is a trilinear form. Alas it is somewhat inconvenient to try and talk about such things strictly in the language of matrix algebra. You can try using partitioned matrices -- it would be a row vector of row vectors of row vectors. (Of course, you could transpose one dimension so that it's a row vector of matrices. But I think that would be awkward...)

P.S. if you insist on directional derivatives being made in the direction of a unit vector, then by $\nabla_v$ I really mean
$$|v| \nabla_{v / |v|}$$​
(Or, in the case where v is zero, I mean the zero function)

3. Jun 19, 2010

### patiobarbecue

my textbook never talks about linear form, bilinear or trilinear form. please give me a good reference. thanks.