The Fundiff command computes functional derivatives of algebraic expressions. If the expression being differentiated contains an integral (maybe a multiple or nested one), Fundiff first attempts to functionally differentiate the integrand, then tries to evaluate the resulting integral, now containing the Dirac delta function or its derivatives. This evaluation, in turn, is carried out by using the integration routines for this problem, which return a result only when the integrand is a polynomial of Dirac functions. Otherwise, the integral is returned unevaluated, after replacing all occurrences of int by Int.

The %Fundiff command is the inert form of Fundiff; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

Since Fundiff is implemented to use Physics/diff, which in turn uses the top level diff command, any special rule for the functional derivative of a user-defined (U.D.) function can be stated by indicating the rule (for the NAME of the U.D. function) to diff in the usual way. In addition, since Physics[diff] recognizes derivatives expressed using the global :-diff or :-Diff syntaxes as mathematically equivalent to those built with the D syntax, the same holds for Fundiff. For the same reason, functional derivatives with respect to indexed fields (functions) where the index represents tensorial properties (spacetime, spinor, or gauge) are computed by Fundiff by using the sum rule for repeated indices, as Physics[diff] does. To define an indexed object as a tensor function, see Define.

When calculating functional derivatives, it sometimes happens that some variables are (constant) Parameters of the theory, and it would be an error to attach functionality to them. To meet this requirement, use the Parameters command of the Physics package, which allows for the declaration of a parameter as a constant parameter in such a way that no functionality can be attached to it.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Coordinates}\left(X\,Y\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\left\{X\,Y\right\}$

$\mathrm{nops}\left(\left[X\right]\right)$

$4$

$X\left[3\right]$

$\mathrm{x3}$

$\mathrm{Define}\left(A\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$\mathrm{ee}≔A\left[\mathrm{\mu },\mathrm{\nu }\right]\left(X\right)$

$\mathrm{ee}≔A_{\mathrm{\mu },\mathrm{\nu }}\left(X\right)$

$\mathrm{Fundiff}\left(\mathrm{ee}\,A\left[\mathrm{\rho },\mathrm{\tau }\right]\left(Y\right)\right)$

${\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\mu }}\mathrm{\rho }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\nu }}\mathrm{\tau }}$

$\mathrm{Fundiff}\left(\mathrm{ee}\,A\left[\mathrm{\rho },\mathrm{\nu }\right]\left(Y\right)\right)$

$4{\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\mu }}\mathrm{\rho }}$

$\mathrm{Fundiff}\left(\mathrm{ee}\,A\left[\mathrm{\mu },\mathrm{\nu }\right]\left(Y\right)\right)$

$16{\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)$

$f\left(X\right)$

$f\left(X\right)$

$'\mathrm{Fundiff}'\left(\,f\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }f\left(Y\right)}\right)f\left(X\right)$

${\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)$

$\mathrm{\%Fundiff}\left(f\left(X\right)\,f\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }f\left(Y\right)}\right)f\left(X\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }f\left(Y\right)}\right)f\left(X\right)$

$\mathrm{value}\left(\right)$

${\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)$

$\mathrm{Setup}\left(\mathrm{differentiationvariables}=X\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\left[\mathrm{differentiationvariables}=\left[X\right]\right]$

$\mathrm{dAlembertian}\left(A\left[\mathrm{\mu },\mathrm{\nu }\right]\left(X\right)\right)$

$\mathrm{\square }\left(A_{\mathrm{\mu },\mathrm{\nu }}\left(X\right)\right)$

$\mathrm{Fundiff}\left(\,A\left[\mathrm{\rho },\mathrm{\sigma }\right]\left(Y\right)\right)$

$\mathrm{\square }\left({\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\mu }}\mathrm{\rho }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\sigma }}}^{\phantom{\mathrm{\nu }}\mathrm{\sigma }}$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(A\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(A\left(X\right)\right)$

$\mathrm{Fundiff}\left(\,A\left(Y\right)\right)$

${\partial }_{\mathrm{\mu }}\left({\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)\right)$

$\mathrm{diff}\left(A\left[\mathrm{\mu },\mathrm{\nu }\right]\left(X\right)\,\mathrm{x0}\right)$

$\frac{\partial }{\partial \mathrm{x4}}{A}_{\mathrm{\mu },\mathrm{\nu }}\left(X\right)$

$\mathrm{Fundiff}\left(\,A\left[\mathrm{\tau },\mathrm{\rho }\right]\left(Y\right)\right)$

${\mathrm{\δ}}^{\left(4\right)}\left(\left[0\,0\,0\,1\right]\,\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\mu }}\mathrm{\tau }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\mathrm{\rho }}$

$\mathrm{Intc}\left(P\left(X\right)\,X\right)$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(X\right){\mathrm{\δ}}^{\left(4\right)}\left(\left[0\,0\,0\,1\right]\,\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\mu }}\mathrm{\tau }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\mathrm{\rho }}\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

$\mathrm{value}\left(\right)$

$-{\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\mu }}\mathrm{\tau }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\mathrm{\rho }}\left(\frac{\partial }{\partial \mathrm{y4}}P\left(Y\right)\right)$

$\mathrm{Intc}\left(\frac{1}{2}{\mathrm{diff}\left(q\left(\mathrm{\tau }\right)\,\mathrm{\tau }\right)}^2-\frac{1}{2}k{q\left(\mathrm{\tau }\right)}^2\,\mathrm{\tau }\right)$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}q\left(\mathrm{\tau }\right)\right)}^2}{2}-\frac{k{q\left(\mathrm{\tau }\right)}^2}{2}\right)\ⅆ\mathrm{\tau }$

$\mathrm{Fundiff}\left(\,q\left(t\right)\right)$

$-kq\left(t\right)-\stackrel{\mathbf{..}}{q}\left(t\right)$

$\mathrm{CompactDisplay}\left(\mathrm{\Phi }\left(X\right)\right)$

$\mathrm{\Phi }\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\Phi }$

$L≔\frac{1}{2}\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)-\frac{m^2}{2}{\mathrm{\Phi }\left(X\right)}^2+\frac{\mathrm{\lambda }}{4}{\mathrm{\Phi }\left(X\right)}^4$

$L≔\frac{{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\Phi }\right)}{2}-\frac{m^2{\mathrm{\Phi }}^2}{2}+\frac{\mathrm{\lambda }{\mathrm{\Phi }}^4}{4}$

$S≔\mathrm{Intc}\left(L\,X\right)$

$S≔{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\Phi }\right)}{2}-\frac{m^2{\mathrm{\Phi }}^2}{2}+\frac{\mathrm{\lambda }{\mathrm{\Phi }}^4}{4}\right)\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

$\mathrm{Fundiff}\left(S\,\mathrm{\Phi }\left(Y\right)\right)$

${\mathrm{\Phi }\left(Y\right)}^3\mathrm{\lambda }-\mathrm{\Phi }\left(Y\right)m^2-\mathrm{\square }\left(\mathrm{\Phi }\left(Y\right)\,\left[Y\right]\right)$

$\mathrm{ans}≔\mathrm{subs}\left(Y=X\,\right)$

$\mathrm{ans}≔{\mathrm{\Phi }}^3\mathrm{\lambda }-\mathrm{\Phi }{m}^2-\mathrm{\square }\left(\mathrm{\Phi }\right)$

$\mathrm{convert}\left(\mathrm{ans}\,\mathrm{diff}\right)$

${\mathrm{\Phi }}^3\mathrm{\lambda }-\mathrm{\Phi }{m}^2+{\mathrm{\Phi }}_{\mathrm{x1},\mathrm{x1}}+{\mathrm{\Phi }}_{\mathrm{x2},\mathrm{x2}}+{\mathrm{\Phi }}_{\mathrm{x3},\mathrm{x3}}-{\mathrm{\Phi }}_{\mathrm{x4},\mathrm{x4}}$

$\mathrm{lprint}\left(\mathrm{ans}\right)$

Phi(X)^3*lambda-Phi(X)*m^2-Physics:-dAlembertian(Phi(X),[X])

$X\left[1\right]$

$\mathrm{x1}$

$X\left[3\right]$

$\mathrm{x3}$

$\mathrm{Define}\left(A\,\mathrm{redo}\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$\mathrm{\%Fundiff}\left(A\left[\mathrm{\mu }\right]\left(X\right)\,A\left[\mathrm{\nu }\right]\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }{A}_{\mathrm{\nu }}\left(Y\right)}\right)A_{\mathrm{\mu }}\left(X\right)$

$\mathrm{value}\left(\right)$

${\mathrm{\δ}}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right){\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

$\mathrm{CompactDisplay}\left(A\left[\mathrm{\mu }\right]\left(X\right)\right)$

$A\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}A$

$F\left[\mathrm{\mu },\mathrm{\nu }\right]≔\mathrm{d\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{\nu }\right]\left(X\right)\right)-\mathrm{d\_}\left[\mathrm{\nu }\right]\left(A\left[\mathrm{\mu }\right]\left(X\right)\right)$

$F_{\mathrm{\mu },\mathrm{\nu }}≔{\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)$

$S≔\mathrm{Intc}\left({F\left[\mathrm{\mu },\mathrm{\nu }\right]}^2\,X\right)$

$S≔{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left({\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)\right)}^2\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

$'\mathrm{Fundiff}'\left(S\,A\left[\mathrm{\rho }\right]\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }{A}_{\mathrm{\rho }}\left(Y\right)}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left({\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)\right)}^2\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

$\mathrm{subs}\left(Y=X\,\right)$

$-2\left(-{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)\right)+\mathrm{\square }\left(A_{\mathrm{\mu }}\right)\right)g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\rho }}}^{\phantom{}\mathrm{\mu },\mathrm{\rho }}+2\left(-\mathrm{\square }\left(A_{\mathrm{\nu }}\right)+{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)\right)g_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\rho }}}^{\phantom{}\mathrm{\nu },\mathrm{\rho }}$

$\mathrm{Simplify}\left(\right)$

$4{\partial }_{\mathrm{\mu }}\left({\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)-4\mathrm{\square }\left(A_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right)$

Cheb-Terrab, E.S. "Maple procedures for partial and functional derivatives." Computer Physics Communications, Vol. 79. (1994): 409-424.