Results 1 to 2 of 2

Math Help - question regarding total derivates

  1. #1
    Junior Member
    Joined
    Oct 2007
    Posts
    43

    question regarding total derivates

    Suppose I have variables x, y and z given parametrically as functions of two variables u and v.

    How can I get the total derivative of z with respect to the variable x, without having to find u and v in function of x and y?

    I know total derivatives can be applied when the different variables are functions of 1 parameter. Here there are two parameters, u and v. How do I proceed?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,356
    Thanks
    36
    Quote Originally Posted by tombrownington View Post
    Suppose I have variables x, y and z given parametrically as functions of two variables u and v.

    How can I get the total derivative of z with respect to the variable x, without having to find u and v in function of x and y?

    I know total derivatives can be applied when the different variables are functions of 1 parameter. Here there are two parameters, u and v. How do I proceed?

    Thanks in advance.
    If x, y, \;\text{and}\; z are given as functions of u\; \text{and}\; v it is often the case that you can't eliminate u\; \text{and}\; v. To calculate the derivative of

    \frac{\partial z}{\partial x}

    you use Jacobians

    \frac{\partial z}{\partial x} = \frac{\partial (z,y)}{\partial (x,y)} = \frac{\partial (z,y)}{\partial (u,v)} \cdot \frac{\partial (u,v)}{\partial (x,y)} = \frac{\partial (z,y)}{\partial (u,v)} / \frac{\partial (x,y)}{\partial (u,v)}

    where the Jacobian is defined as

    \frac{\partial (x,y)}{\partial (u,v)} = \begin{array}{| c c |}x_u & x_v\\y_u&y_v\end{array}

    where subscripts are partial differentiation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using first/second derivates
    Posted in the Calculus Forum
    Replies: 8
    Last Post: September 30th 2009, 06:43 PM
  2. Derivates Question
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 20th 2008, 10:17 AM
  3. limits / derivates question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 24th 2008, 02:14 PM
  4. Natural log and derivates
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 16th 2008, 03:17 PM
  5. derivates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 13th 2006, 01:48 PM

Search Tags


/mathhelpforum @mathhelpforum