1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jun 19, 2010 #1
    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. jcsd
  3. Jun 19, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Actually, f'(x) is not a vector -- it's a linear form. Specifically, it's the linear form with the property that
    [tex]f'(x)(v) = \nabla_v f(x)[/tex]​
    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.
    [tex]f''(x)(v,w) = \nabla_v \nabla_w f(x)[/tex]​
    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. :frown: 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 [itex]\nabla_v[/itex] I really mean
    [tex]|v| \nabla_{v / |v|}[/tex]​
    (Or, in the case where v is zero, I mean the zero function)
  4. Jun 19, 2010 #3
    my textbook never talks about linear form, bilinear or trilinear form. please give me a good reference. thanks.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook