Results 1 to 5 of 5

Math Help - Total Derivative and Orthogonality

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    26

    Total Derivative and Orthogonality




    I did the first part by the definition of total derivative, but can not think of the way to do the second part of the question. Any help please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    If f(x)\in\partial B_r(0) then (f_1(x))^2 + (f_2(x))^2 + \ldots + (f_n(x))^2 = r^2 for all x in U. Differentiate that equation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    26
    Quote Originally Posted by Opalg View Post
    If f(x)\in\partial B_r(0) then (f_1(x))^2 + (f_2(x))^2 + \ldots + (f_n(x))^2 = r^2 for all x in U. Differentiate that equation.
    Thanks for the very quick response.
    Could you please elaborate what you meant by differentiate the equation?
    and how would that help show a vector is orthogonal to (DF)a?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by 6DOM View Post
    Thanks for the very quick response.
    Could you please elaborate what you meant by differentiate the equation?
    and how would that help show a vector is orthogonal to (DF)a?
    I meant differentiate with respect to x, using the chain rule. I suppose I should have used t rather than x, since the question says that f is a function of t. Then \tfrac d{dt}(f_1(t))^2 evaluated at t=a is equal to 2f_1(a)\tfrac {df_1}{dt}(a), and similarly for the other coordinates. The result of differentiating both sides of the equation (f_1(t))^2 + (f_2(t))^2 + \ldots + (f_n(t))^2  = r^2 in that way ought to remind you of an inner product.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2009
    Posts
    26
    Quote Originally Posted by Opalg View Post
    I meant differentiate with respect to x, using the chain rule. I suppose I should have used t rather than x, since the question says that f is a function of t. Then \tfrac d{dt}(f_1(t))^2 evaluated at t=a is equal to 2f_1(a)\tfrac {df_1}{dt}(a), and similarly for the other coordinates. The result of differentiating both sides of the equation (f_1(t))^2 + (f_2(t))^2 + \ldots + (f_n(t))^2  = r^2 in that way ought to remind you of an inner product.
    Understand it perfectly now. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Total derivative of an integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 25th 2011, 06:06 PM
  2. Total derivative of f(A)=A^-1 at I
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 5th 2010, 11:22 PM
  3. Total derivative manipulation
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: May 21st 2009, 03:27 PM
  4. Total derivative
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 16th 2009, 11:52 AM
  5. Total Derivative vs. Directional Derivative
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: May 30th 2008, 08:42 AM

Search Tags


/mathhelpforum @mathhelpforum