Knowlpedia

Definition

Angular average in cylindrical components

The normalized average in the angular variable, with a stated convention for vector components.

For an integrable 2π2\pi-periodic scalar function F(r,θ,z)F(r,\theta,z), its angular average is

⟨F⟩θ(r,z)=12π∫02πF(r,θ,z) dθ.\langle F\rangle_\theta(r,z)=\frac1{2\pi}\int_0^{2\pi}F(r,\theta,z)\,d\theta.

For vector or tensor fields expressed in the moving , a componentwise angular average integrates the cylindrical coefficients separately. This convention must be named because the basis vectors depend on θ\theta.

Why the frame matters

The vector field er(θ)e_r(\theta) has cylindrical coefficients (1,0,0)(1,0,0). Their componentwise average is (1,0,0)(1,0,0), while its average as a Cartesian vector is zero. Similarly, taking an auxiliary mean before evaluating auxiliary variables as functions of (r,θ,z,t)(r,\theta,z,t) need not equal the angular average of the evaluated field.