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Calculus of variations finds stationary points of functionals, often among functions or regular curves.

For fixed $0<\alpha <\pi /2$, the Bessel function of the first kind has an interior nondegenerate stationary point in its cosine integral. The stationary phase method gives the displayed oscillatory leading term and an additive $O(n^{-3/2})$ remainder. The approximation loses uniformity near $\alpha =0$, where the stationary point reaches the endpoint and the curvature vanishes. For negative integer-order arguments, parity supplies a factor $(-1{)}^n$.

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.

The approximate partition function obtained by decoupling each Ising bond has this free energy. The quadratic constant compensates double counting. Its stationary condition is the self-consistency equation; away from stationarity $m$ parametrizes the assumed effective field and need not equal that distribution's mean Ising spin. It agrees at stationary points with the Bragg-Williams free energy of the Ising model, although their off-equilibrium expansions differ.

A stationary point $x_0$ of a smooth phase $\varphi$ is nondegenerate when its Hessian matrix at $x_0$ is an invertible matrix. In one dimension, this condition is ${\varphi }^{\prime \prime }(x_0)\ne 0$.

A stationary point is nonisolated if every neighbourhood contains other stationary points. For a twice differentiable function, a regular smooth curve of stationary points through a point forces its Hessian matrix to annihilate the curve's nonzero tangent: differentiate the vanishing gradient along the curve. Thus the Hessian matrix has a zero eigenvalue there, but that alone need not determine whether the point is a local minimum or local maximum.