curl

Examples of curl in the following topics:

• Curl and Divergence

  • The four most important differential operators are gradient, curl, divergence, and Laplacian.
  • At every point in the field, the curl of that field is represented by a vector.
  • The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
  • A vector field whose curl is zero is called irrotational.
  • The curl is a form of differentiation for vector fields.

• Stokes' Theorem

  • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
  • The Kelvin–Stokes theorem, also known as the curl theorem, is a theorem in vector calculus on $R^3$.
  • Given a vector field, the theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface.

• Conservative Vector Fields

  • Conservative vector fields are also irrotational, meaning that (in three dimensions) they have vanishing curl.
  • Therefore, the curl of a conservative vector field $\mathbf{v}$ is always $0$.
  • A vector field $\mathbf{v}$, whose curl is zero, is called irrotational .
  • However, any simply connected subset that excludes the vortex line $(0,0,z)$ will have zero curl, $\nabla \mathbf{v}=0$.

• Calculus of Vector-Valued Functions

  • Vector functions are used in a number of differential operations, such as gradient (measures the rate and direction of change in a scalar field), curl (measures the tendency of the vector function to rotate about a point in a vector field), and divergence (measures the magnitude of a source at a given point in a vector field).

• Vector Fields

  • Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).