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Use circulation to summarize how much a velocity field runs along an oriented closed curve $C$, such as a circle around a fluid vortex. Its definition is the line integralwhere $d\mathbf{l}$ is tangent to $C$ in the chosen direction. Reversing that direction reverses the sign of circulation. Its units are ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, because velocity is multiplied by a length. Counterclockwise circulation in the horizontal plane is positive when the chosen unit normal vector is $+{\mathbf{e}}_z$, according to the right-hand rule.

For solid-body rotation $\mathbf{u}=(-\Omega y,\Omega x,0)$, a counterclockwise circle of radius $R$ has constant tangential velocity $\Omega R$. Multiplying by its circumference givesFor the potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the same circle instead gives ${\Gamma }_C=(K/R)(2\pi R)=2\pi K$, independent of $R$. Stipulate $\Omega =1{\mathrm{s}}^{-1}$, $R=1\mathrm{m}$ and $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$: both velocity fields have ${\Gamma }_C=2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}\approx 6.283185{\mathrm{m}}^2{\mathrm{s}}^{-1}$, although their vorticities in the fluid are $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$ and zero respectively. On a circle of radius $2\mathrm{m}$, their circulations are $8\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$. The comparison plot in vorticity shows this difference in radius dependence.

The bridge between circulation and vorticity is Stokes theorem:Here the velocity field must be continuously differentiable on a neighborhood of the entire spanning surface $S$ inside the fluid, and its boundary orientation must match the unit normal vector $\mathbf{n}$. For solid-body rotation, the disk is admissible and its surface integral is $2\Omega \pi R^2$. For the potential line vortex, the origin is singular and excluded from the fluid: a disk crossing it is not admissible. On an annulus, Stokes theorem instead gives ${\Gamma }_{R_{\mathrm{out}}}-{\Gamma }_{R_{\mathrm{in}}}=0$, with both circulations here measured counterclockwise; the induced inner boundary direction is clockwise. Nonzero circulation therefore does not imply nonzero local curl in the available fluid. ^[ref]

This is useful because Stokes theorem lets a local vorticity measurement determine a loop's circulation when a suitable surface exists, while Kelvin's circulation theorem tracks circulation along a moving material curve. The beauty is the precise link between a local derivative and a global line integral, with the domain deciding which conclusions are valid.

Let $P\to B$ be a principal bundle with principal connection. Fix a point $b\in B$ and a point $p\in P_b$. The horizontal lift of a piecewise smooth closed curve $\gamma$ based at $b$ starts at $p$ and ends at a unique $p{g}_{\gamma }$. This group element $g_{\gamma }$ is its holonomy relative to $p$.

Changing the starting point to $ph$ changes the group element to $h^{-1}{g}_{\gamma }h$, by equivariance of parallel transport. The conjugacy class is therefore independent of the chosen fiber frame. Transport along a path to another basepoint similarly conjugates the resulting group of holonomy elements.

For a flat principal connection, contractible closed curves have trivial holonomy, but noncontractible closed curves can return a different fiber frame. For example, the U(1) connection $A=\alpha d\theta$ on a trivial circle bundle over a circle has zero curvature form and holonomy $e^{2\pi i\alpha }$ around one circuit. This separates a local curvature condition from a global transport obstruction.

Use irrotational flow to identify where a velocity potential can describe the velocity field locally. An irrotational flow has $\mathbf{\omega }=\nabla \times \mathbf{u}=0$ throughout the fluid region. For a continuously differentiable velocity field, the Poincaré lemma supplies a velocity potential on sufficiently small neighborhoods. A global single-valued velocity potential requires zero circulation around every closed curve; a simply connected domain is a sufficient domain assumption.

The potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$ is an irrotational flow on the punctured plane, or on an annulus $a<r<b$ with $a>0$, even though its pathlines are circular. Every sufficiently small loop bounding a disk in that fluid has zero circulation, by Stokes theorem. A loop winding once counterclockwise around the excluded origin has circulation $2\pi K$. Thus local irrotational flow and nonzero circulation around a hole are compatible: the enclosing disk passes outside the domain of the velocity field.

An irrotational flow is a kinematic condition on a velocity field, independent of whether a particular model neglects viscosity or assumes incompressible flow. Adding incompressible flow makes the local velocity potential satisfy the Laplace equation. This reduces the number of unknowns and makes potential flow a powerful starting point for flow around bodies and for gravity waves. ^[ref]

Knot theory studies closed curves in three-dimensional Euclidean space that cannot intersect themselves, and asks which can be deformed into one another without cutting or passing one strand through another. Its basic object is a knot: a smooth embedding of a circle, considered up to ambient isotopy. Several disjoint knots form a link. More generally, knot theory studies such embeddings in other manifolds and in higher dimensions.

The central surprise is that a flexible circle can carry permanent information about how it sits in space. A trefoil knot and an unknot each consist of just one circle, but no ambient isotopy turns one into the other. A knot diagram translates this spatial question into a finite drawing; Reidemeister moves describe the changes of drawing that preserve the knot. Knot invariants translate it into computations: unequal Alexander polynomials of a knot, Jones polynomials, or knot groups certify different knots. Equal values of a particular knot invariant generally leave the question open; even an isomorphism of knot groups alone is not a complete classification.

An early foundation for working with knot diagrams is Kurt Reidemeister's 1927 paper Elementare Begründung der Knotentheorie, published in volume 5 of Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, pages 24–32.^[ref] Marc Lackenby's authored notes Elementary knot theory, Section 2.4, explain the Reidemeister move theorem and its role in searching for a certificate that two knot diagrams represent the same knot.^[ref]

The chain rule gives $dF(\gamma (t))/dt=F^{\prime }(\gamma (t)){\gamma }^{\prime }(t)=f(\gamma (t)){\gamma }^{\prime }(t)$. The fundamental theorem of calculus therefore yieldsbecause the closed curve has equal endpoints. The existence of a single-valued holomorphic antiderivative on the whole neighborhood is the decisive hypothesis.

A closed curve has $\gamma (a)=\gamma (b)$. A simple curve has no repeated points apart from the permitted equality of its two endpoints when it is closed. A smooth curve, in the regular contour-integration convention, has a continuous nonzero tangent ${\gamma }^{\prime }$; piecewise smooth contours are treated by adding the integrals over their smooth pieces. For a continuous integrand, its contour integral is defined byFor the positively oriented circle, ${\gamma }^{\prime }(t)=ir{e}^{it}$ and $(\gamma (t)-z_0{)}^n=r^n{e}^{int}$. Direct parameter integration therefore givesFor $n\ne -1$, an elementary antiderivative is $e^{i(n+1)t}/[i(n+1)]$ and its endpoint values agree because $n$ is an integer. For $n=-1$, the integrand of the parameter integral is the constant $i$. No integral theorem is needed.

One suitable autonomous planar system isIts equilibrium points are exactly the four combinations $(x,y)=(\pm 1,\pm 1)$. Its Jacobian matrix isThus $(-1,-1)$ and $(1,1)$ have eigenvalues $\pm 2$ and are saddle equilibria. At the other two points the eigenvalues are $\pm 2i$. To establish the nonlinear centre equilibrium classifications, use the first integralIts Hessian matrix is $\mathrm{diag}(2x,-2y)$, a negative-definite matrix at $(-1,1)$ and a positive-definite matrix at $(1,-1)$. Thus each is a strict local extremum, and nearby regular level sets are closed curves supporting periodic orbits, by the planar first integral criterion for a centre.

Immediately to the right of $(-1,1)$, the vector field points down because $x^2-1<0$, so that centre is clockwise. Immediately to the right of $(1,-1)$ it points up because $x^2-1>0$, so that centre is counterclockwise. This verifies both the nonlinear classifications and the prescribed orientations, with exactly four equilibrium points.

For antiplane displacement $u=(0,0,w)$ the only nonzero strain components are $e_{xz}=w_x/2$ and $e_{yz}=w_y/2$. Commutation of mixed partial derivatives gives the necessary local compatibility conditionThe supplied stress is symmetric, independent of $z$ and trace-free. Its first two equilibrium equations are automatic; its third givesaway from the origin. Antiplane shear has zero trace, so isotropic Hooke's law reduces to $\boxed{e=\tau /(2\mu )}$. HenceLocally this integrates to $w=\arg (x+iy)+C$, represented on a suitable branch by $\arctan (y/x)+C$. The strain condition is satisfied at every point of the punctured plane, but the displacement cannot be made globally single-valued there:A gradient of a globally single-valued smooth displacement would instead integrate to zero around any closed curve. Local compatibility is therefore not sufficient on this multiply connected region.

A physical construction is a Volterra screw dislocation: cut the cylinder along a radial half-plane ending on its axis, shift one cut face relative to the other parallel to the axis, and rejoin. The axial mismatch is the Burgers vector. In the given normalization its magnitude is $2\pi$; with a dimensional mismatch $b$ one uses $w=b\arg (x+iy)/(2\pi )$. An intact pristine solid cannot reach this state by a smooth single-valued elastic deformation alone. The axis is a dislocation core where the continuum elastic field is singular and must be regularized by microscopic structure or plastic deformation. On the punctured domain the differential condition holds locally; globally its nonzero circulation detects the defect. In a distributional description, that incompatibility is concentrated on the axis. This is local strain compatibility versus screw-dislocation circulation.

Write the planar velocity field as $\mathbf{u}=(u,v,0)$, with $\mathbf{k}=(0,0,1)$. Expanding the curl givesConsequently, for a piecewise smooth path $\gamma$ from the origin to $\mathbf{x}=(x,y)$,The arbitrary additive constant $\psi (0)$ may be set to zero. The integrand is exactly $\nabla \psi \cdot d\mathbf{r}$, so the fundamental theorem for line integrals proves path independence immediately: the difference between two paths is the line integral of a gradient around a closed curve, which is zero.

One can also see the local integrability condition from Green's theorem:because the given representation enforces incompressibility. This second proof applies whenever the region between the paths lies in the regular fluid domain. The question already assumes a globally defined streamfunction; without that assumption, incompressibility alone would guarantee a global streamfunction on a simply connected domain, but a hole carrying nonzero volume flux could obstruct it.

For smooth inviscid flow of constant mass density under a conservative body force, Kelvin's circulation theorem states that the circulation around any closed material curve is conserved:The constant-mass density assumption is the usual homogeneous incompressible flow setting here. Merely requiring $\nabla \cdot \mathbf{u}=0$ in a fluid of varying mass density would not by itself make the pressure term below a gradient.

Parameterize the material curve by a periodic particle label $s\in [0,1]$, writing ${\mathbf{r}}_t(s,t)=\mathbf{u}(\mathbf{r}(s,t),t)$. Differentiate its line integral, keeping $s$ fixed:The second term vanishes by periodicity. If the conservative body force per unit mass is $-\nabla \Omega$, the Euler momentum equation givesIts line integral around a closed curve is zero, proving the claim. The velocity field must remain smooth on the moving curve, which cannot pass through a line vortex singularity during this argument. No assumption of irrotational flow was needed.

For a simple closed curve in the standard oriented complex plane, positive orientation keeps the bounded interior on the left while traversing the curve. A circle is therefore traversed counterclockwise; reversing direction changes the sign of its contour integral.

The Sagnac effect is a difference in return times, or an associated interference fringe displacement, for light sent in opposite directions around a closed optical path in a rotating apparatus. The light leaves one beam splitter, travels around the same loop in opposite senses, and returns to the rotating beam splitter and readout. For a slowly rotating apparatus in flat spacetime, the leading laboratory arrival-time difference isHere $c$ is the vacuum speed of light, $\mathbf{\Omega }$ is the apparatus's angular velocity relative to an inertial frame, and $\mathbf{A}$ is the vector area oriented by the chosen positive direction around the closed curve $C$. With positive rotation about the upward normal vector, the counterclockwise beam is labelled $+$ and arrives later. Reversing rotation, or reversing the orientation convention alone, changes the sign of $\Delta t$.

The surprise is that a small loop can reveal its rotation by comparing two beams locally, even though each local vacuum speed of light measurement gives $c$. A rotating reference frame is not a single inertial frame: a beam must catch the moving receiver after a full circuit. The Sagnac area law makes this difference a practical way to measure angular velocity, while the Relativistic Sagnac delay on a circular ring distinguishes laboratory coordinate time from the receiver's proper time.

A U(1) connection is a principal connection on a principal bundle with circle group $U(1)$. In a local trivialization it has a real one-form potential $A$, with convention $A\mapsto A+d\chi$ under a gauge transformation $e^{i\chi }$. Its curvature form $F=dA$ is globally defined, although $A$ need not be a global one-form.

For a closed curve $\gamma$, choose a local trivialization of the pulled-back circle bundle along $\gamma$. In unit-charge normalization the holonomy is $\exp (i{\oint }_{\gamma }A)$. Changing that local trivialization changes the integral by $2\pi$ times an integer, leaving the exponential unchanged. When computing with several local potentials, transition phases must be included at chart changes.

On the trivial circle bundle over a circle, $A=\alpha d\theta$ has $F=0$ but holonomy $e^{2\pi i\alpha }$. Thus zero curvature form need not imply trivial holonomy. On any closed oriented surface, the integral $\frac{1}{2\pi }\int F$ is the integral First Chern class, relating the curvature form to the topology of the circle bundle.

Use a velocity potential to reconstruct an irrotational flow from one scalar field. A velocity potential $\varphi (\mathbf{x},t)$ satisfiesIt has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$ and is unchanged as a description of the velocity field when any function of time alone is added. For twice continuously differentiable $\varphi$, the identity curl of a gradient proves zero vorticity. In incompressible flow, taking the divergence also gives ${\nabla }^2\varphi =0$, the Laplace equation. This turns a three-component velocity field into a single harmonic function with appropriate boundary conditions. ^[ref]

For uniform flow $\mathbf{u}=(U,0,0)$, take $\varphi =Ux$. With $U=2\mathrm{m}{\mathrm{s}}^{-1}$, the change from $x=0$ to $x=3\mathrm{m}$ is $\Delta \varphi =6{\mathrm{m}}^2{\mathrm{s}}^{-1}$. By the fundamental theorem for line integrals, the line integral of velocity between these points equals that change along any path in the domain; a closed curve gives zero circulation.

For the potential line vortex, choose a local angle branch in polar coordinates and set $\varphi =K\theta$. Then $u_r={\partial }_r\varphi =0$ and $u_{\theta }=r^{-1}{\partial }_{\theta }\varphi =K/r$. One counterclockwise turn changes the continued velocity potential by $2\pi K$, although the velocity field returns to its original value. With $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$, this change is $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$, exactly the circulation. A full annulus therefore admits local angle branches and a multivalued velocity potential, but no global single-valued velocity potential when $K\ne 0$. Cutting the annulus along a radial line allows a single-valued angle branch on the remaining domain. The obstruction is path independence, not a failure of the local gradient formula.

The winding number records how many net turns a closed piecewise smooth curve makes around a chosen point $w$. Counterclockwise turns contribute positively, clockwise turns negatively. It measures a traversal, including repeated visits; a drawing of the curve's outline alone can lose that information. For a closed curve $\gamma$ avoiding $w$,Equivalently, continuously follow an argument of $\gamma (t)-w$ along one journey and divide its total change by $2\pi$. The start and endpoint agree as complex numbers, so the change is an integer number of full turns. The principal argument at the endpoints cannot retain those turns; a continuous choice along the parametrized path can.^[ref]

For $\gamma (t)=e^{it}$, $0\le t\le 2\pi$, the winding number about zero is one. Traversing the same unit circle twice makes it two; reversing its orientation of a contour makes it negative. The winding number about $w=2$ is zero: the same journey now stays to one side of the point being measured. Joining one counterclockwise loop and one clockwise loop around zero gives net winding number zero even though both loops encircle zero.

The winding number does not change during a homotopy of closed curves avoiding $w$. This is why it is useful: complicated curves can often be deformed to simple ones whose count is obvious. The argument principle transfers that robust geometric integer to a count of zeros of a function and poles. The example in boundary sampling can miss winding shows why finite samples of a curve need control between samples before they can certify the same winding number.