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Derivative of An Inner Product

  1. Jul 13, 2011 #1
    I am trying to take the derivative of an
    inner product (in the most general sense
    over L^2), and was curious if the
    derivative follows the "chain rule" for
    inner products.

    i.e. Does D_y(<f,g>) = <D_y(f),g> + <f,D_y(g)>
    where D_y is the partial derivative w.r.t. y.

    So for example, IT IS TRUE that if f=x*y and g=sin(x*y)
    and the inner product <f,g> = Integral(f#g, w.r.t. x,-Pi,+Pi), f# = complex conj. of f.
    then the equality holds.
    In other words, differentiating w.r.t. y and integrating w.r.t x the forumla holds.
    It seems more trivial if the variable which is being differentiated & integrated is the same.
    But is it true in general?
    What if we are differentiating more abstract inner products (i.e. not necessarily integration).
  2. jcsd
  3. Jul 13, 2011 #2


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    Last edited by a moderator: Apr 26, 2017
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