Source : ct|01.06.13

< Mathematics and physics

Curl of a vector field

The dictionary defines the rotation as "the movement of a body about a point or axis". He defines the adjective rotary as "turning on an axis like a wheel". The word rotationnal does not appear in the commonly used words.
The term "rotationnal" of a vector field is used in mathematics and physics. The English use the term "curl" which means coil, curve, twist, swirl, spiral... What is exactly the definition of this concept and the idea it represents ?…

1) Definition

Consider a vector field
a (x,y,z) = ax(x,y,z) i + ay(x,y,z) j + az(x,y,z) k

We call curl a (in french, rotationnal of vector a) the vector

that we write sometimes, as

with i =(1,0,0), j=(0,1,0), k=(0,0,1), and the nabla operator equal to

2) Interpretation

To illustrate what concretely represents in a point M(x,y,z) the vector rotationnal V of a vector field V(x,y,z), we associate with this point M a surface element infinitely small dS which has an outline dL.
To simplify the calculations, it is assumed that the surface dS is composed of the three following pieces of surfaces :

• the surface element dSi = dydz i, bounded by the contour dLi = (MM5M4M3) in the plane (0,y,z) perpendicular to i
• the surface elementdSj = dxdz j, bounded by the contour dLj = (MM3M2M1) in the plane (0,x,z) perpendicular to j
• the surface elementdSk = dxdy k, bounded by the contour dLk = (MM1M6M5) in the plane (0,x,y) perpendicular tok

The circulation (dC) of the vector V along the outline (dl) surrounding the surface dS is equal to the sum of the circulations of this vector along the paths dLi, dLj and dLk because the circulations of the vector V along the paths MM3, MM5 and MM1 cancel each other. So dS = dSi + dSj + dSk and dC = dCi + dCj + dCk.

The circulation dCi of the vector V, along the path dLi=MM5M4M3 in the plane (0,y,z) perpendicular to i, is equal to dCi = -Vz(x,y,z)dz + Vz(x,y+dy,z)dz - Vy(x,y,z+dz)dy + Vy(x,y,z)dy

The circulation dCj of the vector V, along the path dLj=(MM3M2M1) in the plane (0,x,z) perpendicular to j, is equal to dCj = -Vx(x,y,z)dx + Vx(x,y,z+dz)dx - Vz(x+dx,y,z)dz + Vz(x,y,z)dz

The circulation dCk of the vector V, along the path dLk=(MM1M6M5) in the plane (0,x,y) perpendicular to k, is equal to dCk = -Vx(x,y+dy,z)dx + Vx(x,y,z)dx - Vy(x,y,z)dy + Vy(x+dx,y,z)dy

Circulation along the outlines surrounding the surface elements dydz, dxdz and dxdy

It is thus found that,
dC = dCi+ dCj+dCk = rot V.dS
with

The rotational, of a vector field V at a point M, is different from 0 if the circulation of vector field V (around this point M) is not equal to 0. This condition is satisfied if the vector field rotates (swirls) more or less around this point .

Conversely, if the rotational of a vector field is zero at a point M, this means that the vector field does not rotate around this point M.
Therefore
The rotational vector (at a point M of a vector field V), shows how curls this vector field (more or less) around this point.

Rotational (swirling) of a vector field

Generalization

More generally, we consider any oriented surface S composed of an infinite number of elementary surfaces dS ( infinitely small and oriented). The circulation of vector V along the path (C) around the surface S is equal to the sum of the circulations of this vector along the set of elementary contours because the circulations of the vector V on internal paths cancel each other out (on each internal segments, the circulations take place in two opposite directions and thus cancel each other two by two).

circulation on the contour of a surface

This is why from dC=rot V.dS we deduce naturally

(Stokes Formula)

The Circulation of a vector field V, along the outline of a closed surface S, is equal to the integral of rotationnal of the vector field on the surface. (We take C and S in the direction of the direct trihedral)

direction of the direct trihedral

Rotational (curl) in cylindrical coordinates

The calculation in cylindrical coordinates, of the rotationnal of a vector A at a point M, is done in the same way as in Cartesian coordinates but considering the surface element dS = rdθdz u + drdz v + rdrdθ k around point M(r,θ,z).

• Circulation dCr of A on the elementary oriented surface dSr perpendicular to u :
Aθ(r,θ,z) rdθ - Aθ(r,θ,z+dz) rdθ

Az(r,θ+dθ,z)dz - Az(r,θ,z) dz

• Circulation dCθ of A on the elementary oriented surfacedSθperpendicular to v :
Az(r,θ,z) dz - Az(r+dr,θ,z) dz

Ar(r,θ,z+dz) dr - Ar(r,θ,z) dr

• Circulation dCz of A on the elementary oriented surfacedSz perpendicular to k :
Ar(r,θ,z) dr -Ar(r,θ+dθ,z) dr

Aθ(r+dr,θ,z) (r+dr)dθ - Aθ (r,θ,z) rdθ

We have thus
dC = dCr+ dCθ+dCz égal à

Knowing that dC = dCr+ dCθ+dCz = rot A.dS, and that
dS=rdθdz u + drdz v + rdrdθ k, we obtain

Rotational (curl) in spherical coordinates

The calculation in spherical coordinates, of the rotationnal of a vector A at a point M, is done in the same way as in cartesian or cylindrical coordinates but considering the surface element dS = r²sinθdθdφ u + rsinθdrdφ v + rdrdθ w around point M(r,θ,φ).

• Circulation dCr of A on the elementary oriented surfacedSr perpendicular to u :
Aθ(r,θ,φ) rdθ - Aθ(r,θ,φ+dφ) rdθ

Aφ(r,θ+dθ,φ) rsin(θ+dθ)dφ - Aφ(r,θ,φ) rsinθdφ
= sin(θ+dθ) Aφ(r,θ+dθ,φ) rdφ - sinθ Aφ(r,θ,φ) rdφ

• Circulation dCθ of A on the elementary oriented surface dSθ perpendicular to v :
Ar(r,θ,φ+dφ) dr -Ar(r,θ,φ) dr

Aφ(r,θ,φ) rsinθdφ - Aφ(r+dr,θ,φ) (r+dr)sinθdφ
= rAφ(r,θ,φ)sinθdφ - (r+dr)Aφ(r+dr,θ,φ) sinθdφ

• Circulation dCφ of A on the elementary oriented surfacedSφperpendicular to w :
Ar(r,θ,φ) dr - Ar(r,θ+dθ,φ) dr

Aθ(r+dr,θ,φ) (r+dr)dθ - Aθ(r,θ,φ) rdθ

We have thus,
dC = dCr+ dCθ+dCφ = rot A.dS égal à

with dS=r²sinθdθdφ u + rsinθdrdφ v + rdrdθ w
Therefore

Examples

• Example 1 : Moving particles

Verify the Stokes formula in the case of particles moving along the x axis with a speed V=(xz,0,0) which increases linearly as a function of the abscissa x and of the height z of the particles.

Answer
Using the Cartesian coordinates system, the Stokes formula is applied to a surface S=xz in the y=0 plane.

V = xz i + 0 j + 0 k

The circulation (C) of V along the contour of S (in the sense of the direct trihedral) is reduced to the circulation of vector xz i on AB because the circulations on BC, C0 and 0A are equal to 0.

• Example 2 : Cylinder in rotation about its axis

Calculate the rotationnal of a vector speed (a) of a point M located on a cylinder which rotates about its axis with an angular rotation velocity (ω rad/s)

Answer
Using the cylindrical coordinate system, we have :

The rotationnal of the vector speed S, at a point situated on the cylinder M is equal to 2 times the rotation velocity ω of the cylinder. This example lets us view the concept of a rotational vector field and so understand what it represents concrètement.