Integral theorems · Definition

Curl

multivariable calculusvector calculus

Before you start you need to know

It detects the local swirl of a field; it is the quantity integrated in Stokes' theorem and the one that decides whether a field can be conservative.

Definition

The curl of a vector field \(\mathbf F=\langle P,Q,R\rangle\) is the vector field that measures, at each point, the field's tendency to rotate, with axis and rate of rotation given by its direction and length.

\[\operatorname{curl}\mathbf F=\nabla\times\mathbf F=\langle R_y-Q_z,\;P_z-R_x,\;Q_x-P_y\rangle\]

It is computed as the formal cross product \(\nabla\times\mathbf F\), the determinant with \(\mathbf i,\mathbf j,\mathbf k\) in the first row, \(\partial_x,\partial_y,\partial_z\) in the second and \(P,Q,R\) in the third. Each component is a difference of cross derivatives, e.g. the \(\mathbf k\) one is \(Q_x-P_y\); in the plane, where \(R=0\) and nothing depends on \(z\), only that component remains.

Geometrically, if \(\mathbf F\) is a fluid velocity, a small paddle wheel placed at \(P\) spins fastest when its axis points along \(\operatorname{curl}\mathbf F(P)\), with sense given by the right-hand rule, and the length of the curl is twice the local angular speed. Guiding example: \(\langle -y,x,0\rangle\) has curl \(\langle 0,0,2\rangle\) everywhere, and a rigid body rotating with angular velocity \(\boldsymbol\omega\) has curl \(2\boldsymbol\omega\).

The two basic identities are \(\nabla\times(\nabla f)=\mathbf 0\) and \(\nabla\cdot(\nabla\times\mathbf F)=0\), both consequences of equality of mixed partials. The first gives a practical test: if the curl of a \(C^1\) field on all of \(\mathbb R^3\) is not zero, the field is not conservative. Typical mistake: concluding the converse in regions with holes, where zero curl is not enough.

That is why the curl ties to what came before (conservative fields and line integrals) and to what follows: Stokes' theorem, \(\oint_C\mathbf F\cdot d\mathbf r=\iint_S(\operatorname{curl}\mathbf F)\cdot\mathbf n\,dS\), has Green's theorem as its planar case.

Think about it

If \(\operatorname{curl}\mathbf F=\mathbf 0\) on a solid ball, can you assert \(\mathbf F\) is conservative there?

Yes: a solid ball has no holes, so zero curl is enough for a potential function to exist.

What is the curl of \(\langle -3y,\,3x,\,0\rangle\)?

\(\langle 0,0,6\rangle\): it is a rotation with \(\omega=3\), and the curl is twice the angular velocity.

Exercises 10

  1. Compute \(\operatorname{curl}\mathbf F\) for \(\mathbf F=\langle -y,\,x,\,0\rangle\).
  2. Compute \(\operatorname{curl}\mathbf F\) for \(\mathbf F=\langle yz,\,xz,\,xy\rangle\).
  3. Compute \(\operatorname{curl}\mathbf F\) for \(\mathbf F=\langle xy,\,yz,\,zx\rangle\).
  4. For \(\mathbf F=\langle z,\,x,\,y\rangle\), \(\operatorname{curl}\mathbf F\) is:
  5. Evaluate \(\operatorname{curl}\mathbf F\) at \((1,1,1)\) for \(\mathbf F=\langle x^2y,\,y^2z,\,z^2x\rangle\).
  6. If \(\mathbf F\) is \(C^1\) and conservative on \(\mathbb R^3\), then \(\operatorname{curl}\mathbf F\) is:
  7. For which value of \(k\) does \(\mathbf F=\langle kxy,\,x^2+z,\,y\rangle\) have zero curl?
  8. At a point, \(\operatorname{curl}\mathbf F=\langle 0,\,0,\,3\rangle\). Near that point, the field tends to spin a small object:
  9. The velocity of a rigid body with angular velocity \(\boldsymbol\omega=\langle 1,2,3\rangle\) is \(\mathbf v=\boldsymbol\omega\times\langle x,y,z\rangle\). Its curl is:
  10. For \(\mathbf F=\langle xyz,\,x^2,\,y^2\rangle\), \(\operatorname{div}(\operatorname{curl}\mathbf F)\) equals:

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §16.5
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §16.7
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §4.4