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A flux integral ${\int }_{S}F\cdot ndS$ measures flow through an oriented surface.

The graph $z=f(x,y)$ has area element $\sqrt{1+f_x^2+f_y^2}dxdy$.

A parametrised surface $r(u,v)$ has area element $|r_u\times r_v|dudv$.

The two margin boundaries of a linear support vector machine are the parallel hyperplanes $f(x)=1$ and $f(x)=-1$.

The local Wilsonian effective action superpotential receives no perturbative corrections when supersymmetry is preserved, an infrared cutoff is retained and the same elementary fields are kept. The spurion selection rule for perturbative non-renormalization gives a holomorphy proof. The Kähler potential and physical couplings after wave-function renormalization can change; genuine nonperturbative superpotential terms require a separate analysis.

For canonical global supersymmetry, eliminating the auxiliary fields gives

For solutions ${\phi }_i^{\prime \prime }+q_i{\phi }_i=0$, the mixed WronskiansatisfiesIts endpoint signs between consecutive zeros of a positive ${\phi }_1$ force a zero of ${\phi }_2$ unless $q_1=q_2$.

A toric resolution of singularities is obtained by subdividing a fan until every cone is generated by part of a lattice basis. The resulting smooth toric variety maps properly and birationally to the original one.

The $i$th horn ${\Lambda }_i^n\subseteq {\Delta }^n$ is the union of all codimension-one faces except the face opposite vertex $i$.

An affine scheme is a scheme isomorphic to $\mathrm{Spec}A$ for some commutative ring $A$. Homomorphisms $A\to B$ correspond contravariantly to morphisms $\mathrm{Spec}B\to \mathrm{Spec}A$.

A spectral gap is a positive separation between designated parts of the spectrum of a linear operator. For the power method, the relevant gap separates the modulus of the dominant eigenvalue from the other eigenvalue moduli.

An eigenvalue is dominant when its modulus is greater than or equal to that of every other eigenvalue. It is uniquely dominant when the inequality is strict.

This is the solvability test in Cartan's criterion, distinct from its semisimplicity test.

A complex Lie subalgebra $\mathfrak{h}\subseteq \mathfrak{gl}(V)$ is solvable iffor every $x\in [\mathfrak{h},\mathfrak{h}]$ and $y\in \mathfrak{h}$. For an abstract Lie algebra, this is equivalent to $\kappa (\mathfrak{h},[\mathfrak{h},\mathfrak{h}])=0$.

Every connected solvable affine algebraic subgroup of $\mathrm{GL}(V)$ preserves a complete flag in $V$, equivalently it is conjugate to a subgroup of the upper triangular matrices.

Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.

At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.

The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.

For a nondegenerate alternating matrix $J$, the symplectic Lie algebra is ${\mathfrak{sp}}_{2n}=\{A:AJ+J{A}^T=0\}$.

The special linear Lie algebra ${\mathfrak{sl}}_n$ consists of the trace-zero $n\times n$ matrices with the commutator bracket.

The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type $D_n$.