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An untwisted band is the topological surface obtained from a rectangle by identifying $(0,v)$ with $(1,v)$. It is the Cartesian product of a circle and a closed interval, and is homeomorphic to a closed annulus. A finite portion of the lateral surface of a circular cylinder realizes it in three-dimensional space. Its two boundary circles are $S^1\times \{-1\}$ and $S^1\times \{1\}$, and the matching seam directions make it an orientable surface. The Möbius band changes just one identification sign and loses both this two-component boundary and its orientability.

The upper half-plane is the open setIt contains precisely the complex numbers with positive imaginary part: $1+i$ is inside, $1$ lies on its boundary, and $1-i$ belongs to the lower half-plane. In the complex plane its boundary is the real line; in the Riemann sphere the boundary also includes infinity. It is a convex set, hence a simply connected domain, and the Cayley transform between the half-plane and disk makes it biholomorphic to the unit disc.

With the hyperbolic metric $ds=|dz|/\mathrm{Im}z$, the same domain becomes the Poincaré half-plane model. Its hyperbolic geodesics are vertical lines and upper semicircles whose centres lie on the real line. Under the Cayley transform, these become arcs of circles orthogonal to the unit circle, with diameters as the straight-line cases. ^[ref]

Matching colours identify each hyperbolic geodesic and its image. The vertical line through zero maps to a diameter; the vertical line through one maps to an arc ending at $-i$ and $1$. A semicircle from $-1$ to $1$ also maps to a diameter, whereas the semicircle from $0$ to $2$ maps to a curved arc. Their endpoints lie on the boundary, outside the domains: the endpoints are infinitely far from interior points in hyperbolic distance.

The upper half-plane is useful because its straight boundary and explicit upper half-plane automorphisms make conformal maps and hyperbolic geometry calculable. The picture explains why shortest hyperbolic geodesic routes can look curved in Euclidean coordinates; their lengths are measured with the position-dependent hyperbolic metric.

For historical context, in his letter to Klein of 22 September 1882, Poincaré constructs a quadrilateral bounded by four circles orthogonal to a fundamental circle, and studies transformations generating repeated copies. In modern language, the sides inside that circle are hyperbolic geodesics in a disk model. The letter directly exhibits the connection between transformation groups, geometry and complex functions. ^[ref]

Vector calculus studies differentiation and integration of scalar fields and vector fields. Three operators answer different local questions: the gradient describes change of a scalar value, the divergence describes net outward flow, and the curl describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.

The difference between divergence and curl is especially visible in two smooth vector fields on three-dimensional Euclidean space:At each point they have the same Euclidean norm $\sqrt{x^2+y^2}$, but their directions differ. For $F$, $\nabla \cdot F=2$ and $\nabla \times F=0$. For $G$, $\nabla \cdot G=0$ and $\nabla \times G=(0,0,2)$. One expands radially in each horizontal plane; the other describes rigid rotation.

Around the counterclockwise unit circle in the plane $z=0$, $F$ is normal to the boundary and has zero tangential component. Its outward planar flux integral is $2\pi$, while its line integral around the circle is zero. For $G$ the values are reversed: outward planar flux is zero, and circulation is $2\pi$. The divergence and curl predict exactly which boundary measurement survives.

Why is this useful? A velocity field can move material through a boundary or circulate it within a region. Conservation of mass connects its divergence with density changes through the continuity equation; its curl is vorticity. In electromagnetism, the Maxwell equations use the same operators to connect electric fields, magnetic fields, and their sources. The common mathematical language lets the same integral calculations work across different physical models. ^[ref].

The geometric framework behind these operators is developed in differential forms. The Generalized Stokes theorem unifies the fundamental theorem for line integrals, Green theorem, Stokes theorem, and divergence theorem. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.

Wilson's Vector Analysis, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach vector calculus: learn each operation through a measurement it makes possible. ^[ref].

Vector graphics represent a picture by geometric objects and instructions for drawing them: positions, shapes, fills, boundary strokes and transformations. A circle can be stored as a centre and radius; a rectangle as a corner, width and height; a polygon as an ordered list of vertices. These geometric primitives describe whole objects rather than assigning a colour to every pixel. SVG provides elements for such basic shapes and for paths assembled from line segments and curves.^[ref]

The useful separation is between the stored geometry and the finite output. A circle of radius 2 remains the same geometric description whether rendered across twenty or two thousand output pixels. This is the sense in which vector graphics are resolution independent: increasing the output grid can evaluate the original geometry again. The resulting raster image still has finite spatial resolution, and numerical coordinates have finite precision. Enlarging a previously rendered raster image through interpolation instead works with its existing samples. Neither operation guarantees that arbitrarily thin details remain visible at every size.

An SVG path uses a current point and drawing commands. For example, M 1 4 L 5 4 L 3 1 Z moves to $(1,4)$, draws two line segments, then closes back to the starting point, giving a triangle. Uppercase commands use absolute coordinates; the corresponding lowercase commands use offsets from the current point. Thus M 1 4 l 4 0 l -2 -3 z describes the same triangle. The coordinates belong to the current user Cartesian coordinate system, not necessarily to output pixels.^[ref]

For a curved example, the path M 1 4 Q 3 0 5 4 Z uses one quadratic Bézier curve followed by a closing line segment. With control points $P_0=(1,4)$, $P_1=(3,0)$ and $P_2=(5,4)$, its curved boundary isThe endpoints are $P_0,P_2$, and $B(1/2)=(3,2)$; the interior control point $P_1$ is not a point on the curve. A fill paints the enclosed region, while a stroke paints a band around its boundary with a chosen width. One can use either or both: geometry and paint instructions are distinct.^[ref][ref]

The first illustration uses increasing $y$ downward, following the initial SVG coordinate convention. Its dashed control polygon guides the eye; it is not part of the filled path. Apply the affine mapThis combines horizontal scaling, a shear mapping and a translation. The transformed control points are $(4.5,5)$, $(5.5,1)$ and $(10.5,5)$, giving the path M 4.5 5 Q 5.5 1 10.5 5 Z. Its curved midpoint is $T(3,2)=(6.5,3)$. The affine equivariance of a geometric basis explains why transforming only the control points transforms the entire Bézier curve: its weights sum to one, so the translation is added exactly once. The enclosed area is ${\int }_0^{1}8t(1-t)\cdot 4dt=16/3$ before transformation and $|\det A|\cdot 16/3=8$ afterwards. These checks concern the filled region, excluding the width of any stroke.

An SVG viewBox specifies a rectangle in user coordinates to map into a display region. For example, viewBox="0 0 6 6" in a square viewport of 120 by 120 display units gives a scale of 20 display units per user unit; in a square viewport of 600 by 600 it gives 100. The path's control points need not change. For unequal proportions, preserveAspectRatio controls whether the mapping preserves proportions or stretches the axes separately. These display units need not correspond one for one to physical display pixels.^[ref][ref]

Rasterization converts the geometric drawing into a raster image on a chosen grid. Deciding whether each pixel centre lies inside a filled region is a simple rule, but a moving diagonal boundary can make entire pixels switch abruptly. Antialiasing reduces these sampling artifacts by accounting for a footprint around each sample, rather than treating every pixel as a single inside-or-outside decision. This connects vector graphics to raster sampling and reconstruction and aliasing; a finer grid and a better sampling rule solve different parts of the problem.

For an exact analytical model, fill the region $R=\{(x,y)\in [0,4{]}^2:y\le x+1/4\}$. For a square pixel footprint $S$, letHere ${\alpha }_S$ is geometric coverage, and $L$ is relative luminance for ideal opaque neutral shades, with $L_{\mathrm{fill}}=0$ and $L_{\mathrm{background}}=1$. The intersection and area determine how much of the footprint is covered. This is a specified area-average model, not a claim that every SVG renderer uses exact coverage. The SVG shape-rendering property supplies rendering hints and permits implementation tradeoffs, including disabling antialiasing for some choices.^[ref]

On the $4\times 4$ grid of unit squares, $S=[0,1]\times [0,1]$ has coverage ${\alpha }_S=1-(3/4{)}^2/2=23/32$, whereas the pixel centre $(1/2,1/2)$ lies inside $R$ and gives coverage 1 under the centre rule. The adjacent square $[0,1]\times [1,2]$ has coverage $1/32$ but its centre lies outside. The entire filled area is $16-(15/4{)}^2/2=287/32$. Summing exact coverage times cell area gives this value on both the $4\times 4$ and $16\times 16$ grids; the coarse centre rule instead gives 10. Coverage values are displayed after sRGB encoding of the modeled relative luminance. The thin coloured line is a geometry overlay, not an extra sampled stroke. Enlarging the middle panel by nearest-neighbor interpolation would repeat its sixteen values; re-rendering the geometry produces the 256 independently evaluated footprints in the right panel. Antialiasing reduces abrupt edges but cannot make a finite grid preserve every geometric detail.

The attractive feature of vector graphics is editability at the level of intent: move one control point to bend a boundary, change a circle's radius, or transform a whole diagram while keeping its parts aligned. Logos, technical diagrams and outline fonts benefit because their structure can be expressed with relatively few shapes and curves. A complicated photograph may require enormous geometric detail, so vector graphics do not imply a smaller file than raster graphics. Nor does an SVG container guarantee that every object is vector geometry: its image element can embed or reference a PNG raster image. Transforming that embedded raster image changes its placement and reconstruction, but does not recover missing original detail.^[ref]

In a nonrelativistic elastic collision of incident mass $M$ at speed $u$ with initially stationary mass $m$, the centre-of-momentum frame moves at $V=Mu/(M+m)$. The projectile's outgoing velocity in that frame has fixed magnitude $w=mu/(M+m)$. Its laboratory velocity tips lie on a circle of radius $w$ centred at $V\hat{\mathbf{x}}$. For $M>m$, a tangent from the origin gives the largest laboratory scattering angle, $\sin {\theta }_{\max }=w/V=m/M$. This geometrical result follows from Galilean velocity addition and conservation of energy.

Replace each corner of a polygonal schema by a small closed interval cutting across that corner. The endpoint identifications induced by paired polygon edges glue these intervals into a graph at each vertex equivalence class. This is its vertex link. A circle link gives a disc neighbourhood; a closed interval link gives a half-disc neighbourhood in a surface with boundary.

For a complete pairing of the sides of a single polygon, every link endpoint is paired exactly once. In each vertex equivalence class the link graph is connected, since the corner endpoint identifications generate precisely that class, and all its vertices have degree two. It is therefore a circle. For partial pairings, a connected link can instead be a closed interval when it has two unpaired endpoints. Counting free polygon edges alone does not count boundary components of a surface; their endpoints must first be identified.

For a solid spherical cap of height $0\le h\le 2R$ cut from a sphere of radius $R$,Centre the sphere at the origin with the cap on the side $z\ge R-h$. Its cross-section at height $z$ is a circle of squared radius $R^2-z^2$. A volume integral in cylindrical coordinates givesThe formula includes caps larger than a hemisphere, when $h>R$, and the whole solid sphere, when $h=2R$. For the retained region $z\ge -d$ with $0<d<R$, set $h=R+d$ to obtain $V=\pi (2R^3/3+R^{2}d-d^3/3)$. Multiplying by uniform mass density gives its mass.

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.

Use vorticity to distinguish the local spin of a fluid element from the motion of its center along a pathline. Circular streamlines can occur both with and without vorticity: the two velocity fields below even have the same circulation around the unit circle.

For a continuously differentiable velocity field $\mathbf{u}=(u_x,u_y,u_z)$, vorticity is the curl vectorEach partial derivative divides velocity by distance, so vorticity has units ${\mathrm{s}}^{-1}$. In a planar velocity field independent of $z$, only ${\omega }_z={\partial }_x{u}_y-{\partial }_y{u}_x$ survives. Positive ${\omega }_z$ points along $+{\mathbf{e}}_z$: the right-hand rule associates it with counterclockwise spin viewed from positive $z$ toward the plane. ^[ref]

The local rotational angular velocity is half the vorticity, ${\mathbf{\Omega }}_{\mathrm{spin}}=\mathbf{\omega }/2$. To see the factor, split the velocity gradient $L_{ij}={\partial }_j{u}_i$ into its symmetric rate-of-strain tensor $E=(L+L^T)/2$ and antisymmetric spin tensor $W=(L-L^T)/2$. A nearby displacement $\delta \mathbf{x}$ has relative velocity $L\delta \mathbf{x}$; its rotational part satisfies $W\delta \mathbf{x}=(\mathbf{\omega }/2)\times \delta \mathbf{x}$. This is a proof outline of the local spin interpretation. The rate-of-strain tensor can also change the angle of a particular material line, so that line's angular velocity need not equal $\mathbf{\omega }/2$.

For solid-body rotation, $\mathbf{u}=(-\Omega y,\Omega x,0)$ gives $\mathbf{\omega }=(0,0,2\Omega )$ and ${\mathbf{\Omega }}_{\mathrm{spin}}=(0,0,\Omega )$. Taking $\Omega =1{\mathrm{s}}^{-1}$, the vorticity is $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$, although the angular velocity is only $1{\mathbf{e}}_z{\mathrm{s}}^{-1}$. For the potential line vortex, $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the planar curl is $r^{-1}{\partial }_r(r{u}_{\theta })=r^{-1}{\partial }_{r}K=0$ wherever $r>0$. Its circular pathlines therefore coexist with zero local spin.

This distinction makes vorticity useful for recognizing shear flow as well as swirling motion; taking the curl of the Euler equations also removes a pressure gradient and leads to the vorticity equation. A historical landmark is Hermann von Helmholtz's 1858 paper on fluid vortex motion, an early study devoted to vortex dynamics. ^[ref]

For a Serre fibration $F\to E\to S^n$ with $n\ge 2$, the homological Serre spectral sequence has only columns $0,n$, and its only potentially nonzero differential isUsing an oriented fundamental class of $S^n$ identifies the source and target with $H_i(F)$ and $H_{i+n-1}(F)$. This gives the Wang homomorphism $\Delta$, and the resulting kernel/cokernel filtration pieces splice into the Wang sequence over a sphere.

For a fibration over the circle, the degree-preserving boundary map is instead the monodromy action minus the identity, $T_{\ast }-1:H_i(F)\to H_i(F)$, up to the chosen orientation sign. It arises from the cellular boundary with the fibre-homology local coefficient system. It is already incorporated in the $E_2$ page; there is no additional $d_1$ differential on an $E_2$-starting Serre spectral sequence. The distinction is visible for a product over $S^1$: its monodromy is the identity, so this boundary map is zero. The fundamental group action in the general construction is essential here.^[ref]

The Warsaw circle joins the two path components of a closed topologist's sine curve by an arc attached at their outer endpoints. It is path-connected and has trivial fundamental group, yet admits a continuous map to the circle with no continuous lift to its real-line covering space. Set the map equal to $1$ on the closed topologist's sine curve and wind once on the joining arc. A lift would be constant on each path component of the sine curve, with the same constant on both by continuity, but would change by $1$ along the joining arc. This proves that the map is not null-homotopic, by the homotopy lifting theorem for a covering map. Thus finiteness of the fundamental group alone cannot replace local path connectedness in the usual lifting criterion for a covering space.

A weakly contractible space is a nonempty path-connected topological space $X$ with trivial homotopy groups ${\pi }_n(X,x)$ for every $n\ge 1$. Equivalently, the map $X\to \{\ast \}$ is a weak homotopy equivalence. For a CW complex, the Whitehead theorem then implies that $X$ is contractible.^[ref]

Every contractible space is weakly contractible: a contraction makes every based sphere map nullhomotopic. The real line is a simple example, contracted by $H(x,t)=(1-t)x$. In contrast, the circle fails the test because its fundamental group is $\mathbb{Z}$. For arbitrary topological spaces, weak contractibility alone does not guarantee a continuous contraction; the CW complex hypothesis in the stated implication supplies substantive control over the space.

Over $\mathbb{C}$, the Witt algebra is the Lie algebra of algebraic derivations of the Laurent polynomial ring $\mathbb{C}[z,z^{-1}]$. It has basis ${\ell }_n=-z^{n+1}d/dz$ for $n\in \mathbb{Z}$. The commutator of these differential operators isIndeed, the second-derivative terms cancel in the commutator, leaving the derivative of the coefficient of each operator. The span of ${\ell }_{-1},{\ell }_0,{\ell }_1$ is a copy of the sl2 Lie algebra: take $e={\ell }_{-1}$, $h=2{\ell }_0$, $f=-{\ell }_1$.

The Witt algebra models finite Fourier series of complexified vector fields on the circle. Its nontrivial one-dimensional central extension of a Lie algebra is the Virasoro algebra, whose central term changes the representation theory relevant to conformal field theory.

Let $K\subset S^3$ be a smooth knot and let $m\ne 0$ be an integer. The exterior of its $m$-twist-spun knot in $S^4$ is a fiber bundle over a circle, with fiber the $|m|$-fold cyclic branched covering of $S^3$ along $K$ with an open three-ball removed. For the open complement, use the corresponding branched cover with a point removed. Its monodromy comes from the cyclic deck action.^[ref]

For $m=\pm 1$, the branched cover is $S^3$, the compact fiber is a three-ball, and the resulting two-dimensional surface knot is unknotted. Zero-twist spinning is ordinary spinning and is excluded from this nonzero-twist fibration theorem; a zero-fold branched cover is not its fiber.^[ref]