Ftheta

Ftheta, written as F_θ, denotes the theta component of a vector field F in cylindrical or polar coordinates (r, θ, z). If F is expressed in the cylindrical basis, F = F_r e_r + F_θ e_θ + F_z e_z, where e_r and e_θ are the local radial and tangential unit vectors that depend on θ. The theta component is F_θ = F · e_θ, and in Cartesian form F_θ = -F_x sin θ + F_y cos θ. The unit vectors satisfy e_r = (cos θ, sin θ, 0) and e_θ = (-sin θ, cos θ, 0).

The theta component describes the tangential or rotational part of F about the z-axis. In planar motion, the velocity field can be written as v = ṙ e_r + r θ̇ e_θ, so the theta-velocity component is v_θ = r θ̇.

In vector calculus, F_θ appears in expressions for divergence and curl in cylindrical coordinates. The divergence is ∇·F = (1/r) ∂(r F_r)/∂r + (1/r) ∂F_θ/∂θ + ∂F_z/∂z. The curl components are (∇×F)_r = (1/r) ∂F_z/∂θ − ∂F_θ/∂z, (∇×F)_θ = ∂F_r/∂z − ∂F_z/∂r, and (∇×F)_z = (1/r)[∂(r F_θ)/∂r − ∂F_r/∂θ].

F_θ is widely used in physics and engineering to describe azimuthal flows, rotational fields, and fields with azimuthal variation. In optics and engineering, the term F-Theta (F-θ) appears in the context of scanning systems, where angular deflections are mapped to linear displacements.