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After the sudden parameter change, the functional derivative isEach Cartesian component of each Fourier mode consequently obeysThis is an Ornstein-Uhlenbeck process. The integrating-factor method givesBecause both coefficients are positive, every mode has positive decay rate.

With left functional derivatives for the Grassmann sources, replace fields in the interaction by $\varphi \mapsto i^{-1}\delta /\delta J$, $\psi \mapsto i^{-1}\delta /\delta \bar{\eta }$, and $\bar{\psi }\mapsto -i^{-1}\delta /\delta \eta$. ThusThe ordering displayed fixes the Grassmann signs and makes this an explicit source functional with no dynamical fields.

The polar molecular field is the functional derivative of the free energy with respect to the polar order parameter. In the positive-derivative convention, dissipative dynamics is $D_t\mathbf{p}=-\Gamma \mathbf{h}$ with positive kinetic coefficient. For a quartic local term and an isotropic gradient penalty,Some authors instead define the molecular field with a minus sign, which changes the associated stress and evolution conventions.

The Polchinski equation changes an interaction action with its regulated Gaussian covariance so that the normalized functional integral stays unchanged. In operator notation, ${\dot{e}}^{-S_1}=-\dot{C}:{\delta }^2{e}^{-S_1}/2$. The resulting action flow has a product of first functional derivatives, joining two vertices, and a second derivative, closing two legs into a loop. It generates a Wilsonian effective action, which is not the same functional as the one-particle-irreducible correlation functions encoded by the quantum effective action.

For a condensate with uniform bulk complex field one, its convergent localized momentum isThe subtraction removes the uniform-background phase-gradient contribution. If $\psi -1=O(r^{-1})$ and $\nabla \psi =O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its functional derivative is $\delta p_j/\delta {\psi }^{\ast }=-i{\partial }_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.

A functional derivative of the source term in a Minkowski scalar field path integral inserts $i\varphi (x)$. Time-slicing identifies the resulting product with the time-ordered product of field operators. Dividing by the zero-source vacuum functional removes disconnected vacuum diagrams. The normalized formula has one factor $1/i$ for every insertion.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.