Gradient - Cylindrical and Spherical Coordinates | Cylindrical Spherical Coordinates

Cylindrical and Spherical Coordinates

In cylindrical coordinates, the gradient is given by (Schey 1992, pp. 139–142):

$\nabla f(\rho, \phi, z) = \frac{\partial f}{\partial \rho}\mathbf{e}_\rho+ \frac{1}{\rho}\frac{\partial f}{\partial \phi}\mathbf{e}_\phi+ \frac{\partial f}{\partial z}\mathbf{e}_z$

where ϕ is the azimuthal angle, z is the axial coordinate, and e_ρ, e_φ and e_z are unit vectors pointing along the coordinate directions.

In spherical coordinates (Schey 1992, pp. 139–142):

$\nabla f(r, \theta, \phi) = \frac{\partial f}{\partial r}\mathbf{e}_r+ \frac{1}{r}\frac{\partial f}{\partial \theta}\mathbf{e}_\theta+ \frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi}\mathbf{e}_\phi$

where ϕ is the azimuth angle and θ is the zenith angle.

For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions).