- #1

brydustin

- 205

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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).