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For a polynomial $f$, the distinguished open set $D(f)$ is the locus on which $f$ does not vanish.

A Bingham plastic has $\dot{\gamma }=0$ for $|\tau |<{\tau }_c$ and $\tau ={\tau }_c\mathrm{sgn}\dot{\gamma }+\mu \dot{\gamma }$ after yield.

For an $SU(2)$ connection under a standard trace convention,Its sign depends on orientation and on whether the instanton is self-dual or anti-self-dual.

The persistence length is the decay length of tangent correlations. A common three-dimensional convention is $L_p=A/(k_{B}T)$; a strictly planar chain has tangent-correlation length $2A/(k_{B}T)$.

Power describes how quickly energy is transferred. For mechanical work, the instantaneous power supplied by a force isThe SI unit is $\mathrm{W}=J/s=\mathrm{N}m/s$, distinct from the velocity unit $m/s$. The italic symbol $W$ for work and the upright unit $\mathrm{W}$ for power mean different things. Two processes can transfer the same energy while requiring different powers: doing $12\mathrm{J}$ of work in $2\mathrm{s}$ requires an average $6\mathrm{W}$, whereas doing it in $4\mathrm{s}$ requires an average $3\mathrm{W}$. ^[ref]

For a point mass of constant mass $m$ observed in an inertial frame, the net work along its actual path equals its change in kinetic energy:The middle expression is a line integral; the dot product selects the component of force along the displacement. Positive net work increases speed, negative net work decreases it, and zero net work leaves the endpoint speeds equal. Equal endpoint speeds do not imply equal velocities: a perpendicular force can continually change direction while doing zero work. ^[ref]

The bridge to Newton's second law is short: take the dot product of $m\dot{\mathbf{v}}={\mathbf{F}}_{\mathrm{net}}$ with velocity. The chain rule gives $d(m|\mathbf{v}{|}^2/2)/dt={\mathbf{F}}_{\mathrm{net}}\cdot \mathbf{v}$, equating the rate of change of kinetic energy to instantaneous power. Integrating gives the theorem. No conservative force assumption entered this argument; friction also contributes to the net work.

For the harmonic oscillator with spring stiffness $k=2N/m$, release a mass $m=0.5\mathrm{kg}$ from rest at $x=0.3\mathrm{m}$. The spring exerts $F=-kx$. Its work before the first passage through $x=0$ isThus $m{v}^2/2=0.09\mathrm{J}$ gives speed $0.6m/s$ there. The velocity is negative on this passage because the position is decreasing. The energy calculation alone determines its magnitude, so the direction still requires information about the motion.

For a time-independent potential energy $V$, the conservative force contributes work $-\Delta V$. Separating all other work as $W_{\mathrm{other}}$ gives $\Delta (T+V)=W_{\mathrm{other}}$. Consequently conservation of mechanical energy requires zero net other work; the work-energy theorem itself does not require that conservation. With damping, mechanical energy decreases even though the theorem remains valid. ^[ref]

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

If $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ is continuous and closed, choose a continuous lift $\theta$ such thatThen $\mathrm{wind}(\gamma ,0)=\theta (1)-\theta (0)\in \mathbb{Z}$. This agrees with the contour-integral definition when the path is piecewise smooth.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

Supersymmetry fixes scalar and fermion couplings so that bosonic and fermionic loop contributions to scalar masses have equal quadratic ultraviolet parts with opposite signs.

The divisor class group is the group of Weil divisors modulo principal divisors of rational functions.

A prime Weil divisor is one integral codimension-one closed subscheme. At its generic point, regularity in codimension one supplies a discrete valuation and hence an order of vanishing.

Pitt's theorem implies in particular that every bounded operator $c_0\to {\ell }^1$ is compact. Thus two nonreflexive Banach spaces can have only weakly compact operators between them.

Weakly compact operators form a norm-closed vector subspace of $\mathcal{B}(X,Y)$ and have the ideal property: bounded compositions on either side of a weakly compact operator remain weakly compact.

For Banach spaces $X,Y$ and bounded $T:X\to Y$, the following are equivalent: $T$ is weakly compact; $T^{\ast \ast }(X^{\ast \ast })\subseteq J_Y(Y)$; and $T^{\ast }$ is weakly compact.

A subset of a Banach space is weakly compact when it is compact in the weak topology; it is relatively weakly compact when its weak closure is weakly compact.

The weak-star topology $\sigma (X^{\ast },X)$ is pointwise convergence of functionals on $X$. If $X$ is nonreflexive, it is strictly weaker than the weak topology $\sigma (X^{\ast },X^{\ast \ast })$ on $X^{\ast }$.

In an infinite-dimensional normed space, the weak closure of the unit sphere is the closed unit ball. Every basic weak neighbourhood imposes only finitely many linear conditions, whose common kernel contains a nonzero direction that can move any interior point onto the sphere.

A sequence $(x_n)$ is weakly Cauchy when $(f(x_n))$ is a Cauchy sequence for every continuous linear functional $f$. Its consecutive differences form a weakly null sequence.