Talk:Functional derivative
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> Another definition is in terms of a limit and the Dirac delta function, δ:
> ![\frac{\delta F[\phi(x)]}{\delta \phi(y)}=\lim_{\varepsilon\to 0}\frac{F[\phi(x)+\varepsilon\delta(x-y)]-F[\phi(y)]}{\varepsilon}.](../../../math/8/6/7/86760a7085df0dd8bd482e6d99228a42.png)
But in general, F is a functional which is only defined over continuous/smooth functions. φ(x)+εδ(x-y) does not even count as a function. It's a distribution. Phys 22:03, 3 Dec 2004 (UTC)
Yes, I think that it should be noted that some physicians write the definition as above, but that it isn't a mathematically correct definition. Also, the correct definition (the first one) misses f on the left side. I'll do these corrections. --Md2perpe 22:53, 30 July 2006 (UTC)
[edit] Is this too close to Wolfram's description?
From wikipedia:
"the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function."
From MathWorld (http://mathworld.wolfram.com/FunctionalDerivative.html):
"The functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function."
I don't know how different things need to be in order to avoid looking like someone was just copying their site. http://mathworld.wolfram.com/about/faq.html#copyright --anon
- Thanks. I removed that text, just in case. Oleg Alexandrov (talk) 23:35, 23 October 2005 (UTC)
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- This text was added by User:Kumkee on 13 feb 2004. It was his first (non-anonymous) edit ever; he's only made some half-dozen non-user-space edits since. linas 23:39, 24 October 2005 (UTC)
[edit] Partial Integration?
In the first example in the Examples section, there seems to be a sign error when the "Partial integration of second term" happens. I also think that the text within the math is not good style. Since this is my first time editing a math-related article, I'm not too confident about either of these points. I've changed:
![\begin{matrix} \left\langle \delta F[\rho], \phi \right\rangle & = & \frac{d}{d\epsilon} \left. \int f( \mathbf{r}, \rho + \epsilon \phi, \nabla\rho+\epsilon\nabla\phi ) d^3r \right|_{\epsilon=0} \\ & = & \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d^3r \\ & = & [\mbox{partial integration of second term}] \\ & = & \int \left( \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \phi \right) d^3r \\ & = & \left\langle \frac{\partial f}{\partial\rho} + \nabla \cdot \frac{\partial f}{\partial\nabla\rho}, \phi \right\rangle \end{matrix}](../../../math/d/3/f/d3f6232c1082f9c2079924d539ee4469.png)
to
![\begin{matrix} \left\langle \delta F[\rho], \phi \right\rangle & = & \frac{d}{d\epsilon} \left. \int f( \mathbf{r}, \rho + \epsilon \phi, \nabla\rho+\epsilon\nabla\phi )\, d^3r \right|_{\epsilon=0} \\ & = & \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d^3r \\ & = & \int \left( \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \left[ \frac{\partial f}{\partial\nabla\rho} \phi \right] - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\ & = & \int \left( \frac{\partial f}{\partial\rho} \phi - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\ & = & \left\langle \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}\,, \phi \right\rangle \end{matrix}](../../../math/2/6/e/26e80119d43df6139ac832532cf1b46c.png)
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Eubene 22:13, 18 September 2006 (UTC)
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- You're correct; I had made an error. Good change! Md2perpe 20:58, 26 September 2006 (UTC)
