vector ﬁelds - sorbonne-universite.fr

Vector fields

Chantal Oberson Ausoni

6.8.2014

ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 1

Vector fields: Outline1. VECTOR FIELDS: MOTIVATION, DEFINITION

2. INTEGRAL LINES, SINGULARITIES, VECTOR FIELD TOPOLOGY

• Integral lines, phase portrait, separatrices, flow• Classification of the integral lines• Hyperbolic singularities and linearization• 2D classification of hyperbolic stationary points• 3D classification of hyperbolic stationary points• Index of a singularity• Poincaré-Hopf theorem• Higher order singularities• Orbits• Vector field topology

3. DIVERGENCE, CURL, HELMOLTZ-HODGE DECOMPOSITION

4. VISUALIZATION OF VECTOR FIELDS: SMALL CATALOGUE

• Sampled vector field• Streamlines, streamsurfaces, streamribbons• The problem of seeding• Line integral convolution


Vector fields: motivation, definition

Vector fields: why, where?

A vector field arises in a situation where,for some reason, there is a direction anda magnitude assigned to each point of thespace or of a surface, typically examples arefluid dynamics, wheather prediction, ...A classical example would be to representthe velocity of the wind with a vector.



Examples of vector fields



Vector fields: definitionsDEFINITION 1. A vector field is a smooth function f : U ⇢ Rn �! Rn. In this frame, thevector field is steady: it does not depend on the time.EXAMPLE 1. Any problem implying the gradient of a scalar function defines a vector field; if

g : U ⇢ Rn �! R is a smooth function, then the gradient rg =

0

BBBBB@

@g

@x1@g

@x2· · ·@g

@x

n

1

CCCCCAis a vector field.

DEFINITION 2. Let S be a surface. A vector field on S is an assignment to each point p 2 S

of a tangent vector X(p) at p, such that these vectors vary in a smooth manner.

On a surface patch given by a parametrization f , we can express the vector field by theexpression

X = X1(u, v)@f

@x

(u, v) +X2(u, v)@f

@x

(u, v)

where X1 and X2 are smooth functions of the parameters (u, v).


Integral lines, singularities, vector field topology

Correspondance between vector fields and ODEsDEFINITION 3. An autonomous, ordinary differential equation, defined on an open setU ⇢ Rn, is an equation of the form: dx/dt = f(x), where x : I �! U denotes anunknown curve, f : U ⇢ Rn �! Rn is C1.THEOREM 1. [Existence and uniqueness theorem for solutions of autonomous ODEs]

Given any point x0 2 U , there exists an interval I containing 0, and a solution x : I �! U ,satisfying x

0(t) = f(x(t)) and the condition x(0) = x0 (called initial condition), and itis unique on the interval I.

In a mathematical sense, vector fields and differential equations may be consideredto be the same thing: Given a vector field defined by a function f : U �! Rn,one may write the differential equation given by dx/dt = f(x). And conversely,given a differential equation dx/dt = f(x), one may extract the vector field f(x).

REMARKS. • Think of f(x) as representing the velocity of a steady wind at the pointx. Then the solution curve x(t) represents the hypothetical trajectory of a masslessparticle released at time t = 0 at the point x0.

• Non autonomous differential equations dx/dt = f(x, t) correspond in the same wayto unsteady vector fields. Example: the real wind is varying with time.



Integral lines, phase portrait, separatricesDEFINITION 4. • Maximal solutions of

these ODEs are called integral lines,trajectories or streamlines, they aretangent in every point to the vector field.

• Given an initial condition x(t0) = x0,the streamline is given by

x(t) = x0 +Z

t

t0f(x(u))du.

• The set of integral curves is called phaseportrait.

• It gives a partition of the domain in re-gions of similar behaviour, where partic-ular integral lines (the separatrices) makethe separation between the regions.

REMARK 1. Let g : U ⇢ Rn �! R be a function, the level lines g(�(t)) = c are orthogonalto the gradient rg, hence to the streamlines defined by this vector field.



FlowWe define a mapping ', called flow, giving all trajectories at the same time:

• �(x, t) is the trajectory from point x (in time 0, the position is x: �(x,0) = x).

• �(�(x, s), t) = �(x, s+ t), for all s and t where it is defined (Consistency condition).REMARKS. • The existence and uniqueness theorem for ODEs tells that this function exists

and is C1.

• We can also consider the family of transformations: �t

: U �! U , obtained by keepingt constant. The consistency is then stated as �

s

� �

t

⌘ �

s+t

, for all s and t where it isdefined.



Classification of the integral lines• A regular trajectory never returns to where it has been.

• If the curve x(t) has a period t0, i.e., x(t+ t0) = x(t) for all t, it is a simple closedcurve and is called closed orbit.

• If the trajectory stays put in p, i.e., x(t) = p for all t, p is a stationary point, criticalpoint or singularity. It means that v vanishes in p.

Regular trajectory, two kind of stationary points, orbit [Frey]



Counter-example of the uniquenessEXAMPLE 2. Let us consider the vector field v(x, y) = (1,3 y

2/3). The derivative @v

@y

(x, y) =

2 y

�1/3 does not exist in zero, meaning that the existence and unicity theorem cannot beapplied in this context.Actually, for any point (x,0) of the x-axis, there are two trajectories meeting:

u1(t) = (x+ t,0) and u2(t) = (x+ t, t

3).



The Jacobian matrix (reminder)The Jacobian matrix Dv(p) =

✓@v

i

@x

j

◆is the matrix of all first-order partial derivatives of v:

• in 2D,v(x1, x2) = (v1(x1, x2), v2(x1, x2)):

Dv(p) =

0

@@v1@x1

(p) @v1@x2

(p)@v2@x1

(p) @v2@x2

(p)

1

A

• in 3D,v(x1, x2, x3) = (v1(x1, x2, x3), v2(x1, x2, x3), v3(x1, x2, x3)):

Dv(p) =

0

BBB@

@v1@x1

(p) @v1@x2

(p) @v1@x3

(p)@v2@x1

(p) @v2@x2

(p) @v2@x3

(p)@v3@x1

(p) @v3@x2

(p) @v3@x3

(p)

1

CCCA



Eigenvalues/eigenvectors (reminder)DEFINITION 5. A matrix A having a special vector u 6= 0 such that Au = �u is said to havean eigenvector u. The scaling factor � is called eigenvalue of A for u.

The eigenvalue equation for a matrix A is Av � �v = 0.

For a 3 ⇥ 3-matrix, it is equivalent to (A � �I3)v = 0, where I3 is the 3 ⇥ 3 identitymatrix. It is a fundamental result of linear algebra that an equation Mv = 0 has a non-zerosolution v if, and only if, the determinant det(M) of the matrix M is zero. It follows thatthe eigenvalues of A are precisely the real numbers � that satisfy the equation

c

A

(�) = det(A� �I3) = 0.

This polynomial is called the characteristic polynomial of A. Since the characteristic poly-

nomial always has three roots in C, there are always three eigenvalues.



Eigenvalues/Eigenvectors (symmetric matrix)When A is a 3 ⇥ 3 symmetric matrix, i.e., when A

T = A, there are 3 real eigenvalues,say �1,�2,�3. Furthermore, all corresponding eigenvectors u1, u2, u3 are orthogonal. Ifwe normalize them and build the matrix U =

⇣u1 u2 u3

⌘, U will be orthonormal:

U · UT = I3. We have the equation0

B@�1 0 00 �2 00 0 �3

1

CA = U

T

AU

(corresponding to a change of basis, for an linear map having matrix A in the standard basis,to a new basis constituted of the eigenvectors).



Eigenvalues/Eigenvectors in 3D(Example)

EXAMPLE 3. For the matrix

0

B@1 2 0�1 4 00 0 3

1

CA,

• the characteristic polynomial is the determinant of the matrix

0

B@1� � 2 0�1 4� � 00 0 3� �

1

CA:

c

A

(�) = (3� �)[(1� �)(4� �) + 2] = (3� �)(3� �)(2� �)

• the eigenvalues are the numbers 2,3,3

• the vectors⇣1 1 0

⌘T

and⇣0 0 1

⌘T

are eigenvectors for the value 3

• the vector⇣2 1 0

⌘T

is an eigenvector for the value 2.



Hyperbolic singularities and linearizationNear a singularity v(p0) = 0, one can consider a local linearization (Taylor):

v(p) = Dv(p0)(p� p0) + 0((p� p0)2)

DEFINITION 6. We call the singularity p0 non-degenerate if the Jacobian matrix Dv(p0) isinvertible, i.e. if its eigenvalues are all non-zero.

A non-degenerate singularity is isolated, as a consequence of the local inversion theorem:on a small neighborhood of p0, v is a bijection, hence v(p) 6= 0.DEFINITION 7. A singularity p0 is called hyperbolic (linear) singularity if the real parts of theeigenvalues of Dv(p0) are all non-zero.

Hyperbolic singularities are stable (i.e., a local perturbation doesn’t change the topology ofthe trajectories near the point). Singularities are nearly always hyperbolic.

The eigenvalues of Dv(p0) allow a classification of the hyperbolic singularities.This classification corresponds to a geometric description of the behaviour of inte-gral lines in the vicinity of the singularity.


Integral lines, singularities, vector field topology2D classification of hyperbolic stationary points

Let us consider an hyperbolic singularity and the Jacobian matrix in this point. The eigen-values of the Jacobian are the roots of the characteristic polynomials and are either both realor both complex (in this case, they are conjugates):

1. Both eigenvalues are real:• both eigenvalues are positive: it is a source (repelling node)

• both eigenvalues are negative: it is a sink (attracting node)

• the eigenvalues have opposite signs: it is a saddle

2. Both eigenvalues are complex: �1 = a+ ib and �2 = a� ib

• a is positive: it is a spiral source (repelling focus)

• a is negative: it is a spiral sink (attracting focus)

The 2D hyperbolic singularities, and the non-hyperbolic center [Theisel]ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 16


3D classification of hyperbolic stationary pointsLet us consider an hyperbolic singularity and the Jacobian matrix in this point. Either thethree roots are real, or there is a real root and a pair of complex conjugate ones.

1. All three eigenvalues are real:

• all eigenvalues are positive: it is a source (repelling node)

• all eigenvalues are negative: it is a sink (attracting node)

• one eigenvalue is positive, two are negative: it is a saddle (1 dimension out, 2 dimen-sions in)

• one eigenvalue is negative, two are positive: it is a saddle (2 dimensions out, 1 di-mension in)

2. One real eigenvalue �1 and two complex ones: �2 = a+ ib and �3 = a� ib

• �1 is positive:

� if a is positive: spiral source

� if a is negative: spiral saddle (2 dimensions in, 1 dimension out)

• �1 is negative:

� if a is positive: spiral saddle (2 dimensions out, 1 dimension in)

� if a is negative: spiral sink



3D classification of hyperbolic stationary points

The 3D hyperbolic singularities [Scheuermann]



Counter-example for the linearizationEXAMPLE 4. If the singularity is not hyperbolic, there is no stability, and the linearization is

not valid. Consider the ODE

8<

:

dx

dt

= �y � x (x2 + y

2)dy

dt

= x� y (x2 + y

2).

• The point (0,0) is a singularity.

• The Jacobian matrix is �3x2 � y

2 �1� 2x y

1� 2x y �x

2 � 3y2

!

.

• In (0,0), the Jacobian matrix is 0 �11 0

!

, with eigenval-

ues ±i. As a consequence, the singularity is not hyperbolic.

• The linearization of the system around (0,0) is

8<

:

dx

dt

= �y

dy

dt

= x

The singularity of the linearized system is a centre.

• On the figure, we see that the singularity is not a centre but a sink.


Integral lines, singularities, vector field topologyIndex of an isolated singularity of a vector field

This (not completely formal) definition uses the angular variation of the vector field aroundthe singularity. It is useful to classify higher order singularities.DEFINITION 8. Let v = (v1, v2) be a vector field on a surface S, x an isolated singularity ofv, ✓ be the angular coordinate of the vector field, the Poincaré index of v in x is the integral

ind

x,v

=1

2⇡

Z

�

d✓ =1

2⇡

Z

�

v1dv2 � v2dv1

v

21 + v

22

where � is a simple closed curve on S, containing x as only singularity of v and makingone turn counterclockwise around it.

The first field makes one counterclockwise turn along the curve, so the star has index +1.The second one makes a clockwise turn along the curve, the saddle point has index -1.[Scheuermann]REMARKS. • ✓ is the arctangent of (v2

v1)

• We see that it measures the number of revolutions of v along the curve �.

• It can be generalized in higher dimensions: for a singularity in a 3D-vector field, forexample, we would take a sphere, instead of a closed curve, around it.



Example of the indexEXAMPLE 5. Planar vector fields

We can also represent all (normalized) vectorsalong the curve from the origin, hence defin-ing a path '(t) = v(�(t))

kv(�(t))k on S

1 and countthe number of turns made by this path aroundthe origin.

On the left the vector field v in the plane, on the right the normalized v(x)kv(x)k in S

1.[Mann]



Poincaré-Hopf theoremTHEOREM 2. (Poincaré-Hopf). Let M be a compact oriented surface and v : M �! TM asmooth vector field with isolated zeros. The sum of the indices at the zeros equals the Eulercharacteristic of M :

X

x zero of vind

x,v

= �(M)

In particular, if the Euler characteristic is non-zero, thenthe vector field must have singularities. This result,when stated for the sphere (� = 2), is called hairyball theorem.If you consider the atmospheric wind circulation tohappen in the same air layer around the earth and ne-glect the non-tangential component of it, it means thatthere is at any time a cyclone somewhere.

”You cannot comb a coconut. Can you comb a doughnut?”



Higher order critical points, their indicesA singular point for which the Jacobian matrix is degenerate is said higher order singular-ity. Such singularities can be classified through their index or described by the successionof sectors.

Higher order singularities, hyperbolic, elliptic and parabolic sectors [Scheuermann, Theisel]

If the goal is to visualize singularities of the right index, complex analysis can be used:

Complex functions presenting singularities of index +1 and +2 [Scheuermann]



OrbitsWe consider the 3D-case: the 2D case is simpler and follows from it.

Considering an orbit C and a point p 2 C we can define its associated Poincaré map:choosing around p a small disk D as a local section of the vector field (meaning that thevector field is nowhere tangent to this disk), there exists a smaller disk D0 ⇢ D with:

for every q 2 D0, you can define a smallest value t(q), such that 't(q)(q) = q

0 isin D.

DEFINITION 9. The mapping q 7! '

t(q)(q) is called Poincaré map.

An orbit is said hyperbolic if the eigenvalues of the Jacobian matrix are off the unit circleof the complex plane. Orbits are structurally stable iff they are hyperbolic. In this case,the Jacobian matrix of the Poincaré map contains all necessary information to classify thedifferent orbits.



Classification of orbits

Closed orbits on surfaces, sourcesand sinks orbits([Chen])

In 2D, the Jacobian matrix is 1 ⇥ 1,there is only one root �:

1. if |�| > 1, it is a source closedorbit

2. if |�| < 1, it is a sink closed or-bit

In 3D,1. both roots �1 and �2 are real: both eigenvalues

have the same sign (since Euclidean space is ori-entable).• |�1| > 1, |�2| > 1: source closed orbit• |�1| < 1, |�2| < 1: sink closed orbit• |�1| > 1, |�2| < 1:

� both positive: saddle closed orbit� both negative: twisted saddle closed orbit

2. both roots �1 and �2 are complex conjugates:• |�1| > 1, |�2| > 1: spiral source closed orbit• |�1| < 1, |�2| < 1: spiral sink closed orbit

REMARK 2. Closed orbits can be knotted or even linked with other orbits.



Vector field topologyThe vector field topology (or topological skeleton) is the giving of

• the critical points (with type)

• the separatrices

• the orbits (with type) of the given vector field.

The separatrices partition the vector field in areas of similar behavior.

Topological skeleton of a vector field


Integral lines, singularities, vector field topologyVector field topology: an example

Vector field topology: two sources, one saddle [Scheuermann]



Vector field topology: non-linear topology

Non-linear topology [Scheuermann]

See the higher-order singularities.


Divergence, curl, Helmoltz-Hodge decomposition

Divergence and volume-preserving vector fieldsDEFINITION 10. A vector field is volume-preserving if its flow '

t

carries any open set S inits domain to a set '

t

(S) of the same volume, for all times t.DEFINITION 11. The divergence of a vector field v of coordinates (v1, v2, v3) is a scalar-valued function:

div v =@v1

@x

+@v2

@y

+@v3

@z

.

The notation div v = r · v is often used.

The condition for a vector field v to preserve volume is that its divergence div v isequal to zero.

For this reason, the term divergence free is used as a synonym for volume-preserving. Insuch a vector field, no open set may ever reduce its volume, whether in positive or negativetime; as a consequence, there are no sinks and no sources.



CurlIn vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a3-dimensional vector field. It cannot be extended easily to higher dimensions.DEFINITION 12. If you have a 3D vector field v with cartesian coordinates (v1, v2, v3), itscurl is defined as

curl v =

@v3

@y

�@v2

@z

!

e1 +✓@v1

@z

�@v3

@x

◆e2 +

@v2

@x

�@v1

@y

!

e3.

The curl of v is often written r⇥ v.

Intuitive explanation: If you imagine that your vector field is the velocity field for a fluid, inevery point you can imagine a small ball, that the fluid brings to rotate. The direction of thecurl is the axis of this rotation. The magnitude of the curl is twice the angular speed.



Helmoltz-Hodge decomposition for vector fieldsTHEOREM 3. (Helmholtz-Hodge Decomposition) A smooth vector field v, defined on abounded or an unbounded domain, can be uniquely decomposed into three components:

1. an irrotational component d,

2. an incompressible component r,

3. an harmonic component h, representing the translation part.

The components d and r can be calculated as the gradient of a scalar potential D and thecurl of a vector potential R, respectively,

v = rD +r⇥R+ h = d+ r + h.

REMARK 3. The harmonic part being at thesame time irrotational and incompressible canbe put together with the first or the secondterm, giving a version of the decompositionwith only two parts.

Example of a HHD decomposition [Polthier]:d, with scalar potential D, r with scalar po-tential R (2D case), h, and the original vectorfield v.


Visualization of vector fields: small catalogue

Sampled vector fieldSuppose that we have a sampled vector field on a triangulation. Using interpolation of thevector field inside the triangles, on can draw streamlines and even sketch the vector fieldtopology. It is clear that both the streamlines and the vector field topology depend on thetype of interpolation used, sometimes delivering wrong topologies:

Original vector field, linearly interpolated vector field, Powell-Sabin interpolated vector field, with the corre-

sponding vector field topologies [Scheuermann]



Streamlines, streamribbons• The streamsurfaces are sometimes used to visualize the flow in 3D. They are formed of

the streamlines rooted at every point along a line segment or a curve.

• The streamribbons are generated from the tangent and normal unit vectors to the stream-line. The twists of a streamribbon reflect the properties of the local flow.

Hedgehog versus streamlines [Amebarki], streamlines versus streamtapes [Chen]



The problem of seedingThere should not no be empty region, and no crowdedone!Different strategies are used: seeds can be distributedevenly, streamlines can be distributed evenly, seeds canbe distributed according to an entropy (reflecting theareas carrying more “information”)...

Different seeding strategies: grid-seeded versus optimized [Turk]



Line integral convolutionA random noise pattern is chosen and a convolution ismade over every streamline.Let v be a 2D-vector field, x a point,�

x

: I = [�1,1] �! R2, a unit-speed parametriza-tion of a streamline such that �

x

(0) = x. The LIC at xis defined as:

Im(x) =Z

I

⌦(s)N(�x

(s))ds.

where ⌦(s) is a Gaussian (for example) convolutionkernel on I, N(x) is the noise image in x.The method is applicable to surfaces in higher dimensions, using multidimensional noisefields.



Example: different strategies for the same vector field

[Scheuermann]


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