Results 1 to 4 of 4

Math Help - vector-valued function

  1. #1
    Junior Member cinder's Avatar
    Joined
    Feb 2006
    Posts
    60

    vector-valued function

    Find the open interval(s) on which the curve given by the vector-valued function is smooth.

    r(t)=t^2i+t^3j

    If I follow the book correctly, I get the derivative which is r'(t)=2ti+3t^2j and then after that I have no idea.

    In the example problem it shows r(t)=0i+0j and list some intervals, but I'm not sure what they're doing or what's going on.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by cinder View Post
    Find the open interval(s) on which the curve given by the vector-valued function is smooth.

    r(t)=t^2i+t^3j
    A vector function is smooth iff it is differenciable and \bold{r}'(t)\not = 0.

    Thus,
    \bold{r}'(t) = 2t\bold{i}+3t^2\bold{j}
    It can only be be zero if t=0. Thus any open interval NOT containing zero makes this smooth.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,052
    Thanks
    368
    Awards
    1
    Quote Originally Posted by ThePerfectHacker View Post
    A vector function is smooth iff it is differenciable and \bold{r}'(t)\not = 0.

    Thus,
    \bold{r}'(t) = 2t\bold{i}+3t^2\bold{j}
    It can only be be zero if t=0. Thus any open interval NOT containing zero makes this smooth.
    Why would smoothness require that \bold{r}'(t)\not = 0?

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by topsquark View Post
    Why would smoothness require that \bold{r}'(t)\not = 0?

    -Dan
    Because the intuitive meaning is that we can draw a tangent vector at every point. Now what type of tangent is a zero vector?


    I believe the mathematical meaning is that we end up dividing by \bold{r}' within the computation. And to keep it valid we require vectors being smooth for certain theorerms.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Graphing a vector valued function
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 15th 2011, 09:31 AM
  2. Vector Valued Function Problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 18th 2009, 02:45 PM
  3. Limit of Vector Valued Function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 17th 2009, 03:58 AM
  4. Vector-Valued Function
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 10th 2008, 05:50 PM
  5. vector valued function
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 22nd 2007, 04:54 PM

Search Tags


/mathhelpforum @mathhelpforum