Results 1 to 2 of 2

Math Help - square of differential operator

  1. #1
    Junior Member
    Joined
    Nov 2009
    Posts
    54

    square of differential operator

    Can differential operators be squared algebraically in the usual way? What would this even mean? Is this just the differential operator applied twice consecutively?

    For example, does it make sense to compute (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y})^2 = (\frac{\partial}{\partial x})^2 -2i\frac{\partial}{\partial x}\frac{\partial}{\partial y}-(\frac{\partial}{\partial y})^2? If so, what is to be done with, say, (\frac{\partial}{\partial x})^2? Is this just \frac{\partial ^2}{\partial x^2}?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,410
    Thanks
    1328
    Quote Originally Posted by cribby View Post
    Can differential operators be squared algebraically in the usual way? What would this even mean? Is this just the differential operator applied twice consecutively?

    For example, does it make sense to compute (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y})^2 = (\frac{\partial}{\partial x})^2 -2i\frac{\partial}{\partial x}\frac{\partial}{\partial y}-(\frac{\partial}{\partial y})^2? If so, what is to be done with, say, (\frac{\partial}{\partial x})^2? Is this just \frac{\partial ^2}{\partial x^2}?
    Yes, in "operator" notation, the basic operation is composition so \left(\frac{\partial }{\partial x}- i\frac{\partial }{\partial y}\right)^2 = \frac{\partial^2}{\partial x^2}- 2i\frac{\partial^2}{\partial x\partial y}- \frac{\partial^2}{\partial y^2}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differential Operator Norm
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 29th 2011, 06:13 AM
  2. Inverse Differential Operator ..
    Posted in the Differential Equations Forum
    Replies: 10
    Last Post: December 29th 2010, 04:08 AM
  3. Adjoint of a differential operator
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 23rd 2010, 05:16 AM
  4. Is this wrong? (differential operator)
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: June 4th 2010, 08:11 PM
  5. Differential Operator Notation Question
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 12th 2010, 06:49 PM

Search Tags


/mathhelpforum @mathhelpforum