Newton's Second Law can be written as follows if only the pressure gradient, gravity, and friction act on an air parcel, where bold symbols are vectors:
Here U is the velocity field of the air, Ω is the angular velocity vector of the planet, ρ is the density of the air, P is the air pressure, F is the friction, g is the acceleration vector due to gravity and  is the material derivative.
Locally this can be expanded in Cartesian coordinates, with a positive u representing an eastward direction and a positive v representing a northward direction. Neglecting friction and vertical motion, as justified by the Taylor–Proudman theorem, we have:
With  the Coriolis parameter (approximately , varying with latitude).
Assuming geostrophic balance, the system is stationary and the first two equations become:
By substituting using the third equation above, we have:
with Z the height of the constant pressure surface (geopotential height), satisfying
This leads us to the following result for the geostrophic wind components (u, v):
The validity of this approximation depends on the local Rossby number. It is invalid at the equator, because f is equal to zero there, and therefore generally not used in the tropics.
Other variants of the equation are possible; for example, the geostrophic wind vector can be expressed in terms of the gradient of the geopotential Φ on a surface of constant pressure: