In order for a map projection of the sphere to be equal-area, its generating formulae must meet this
Cauchy-Riemann-like condition: :\frac{\partial y}{\partial \varphi} \cdot \frac{\partial x}{\partial \lambda} - \frac{\partial y}{\partial \lambda} \cdot \frac{\partial x}{\partial \varphi} = s \cdot \cos \varphi where s is constant throughout the map. Here, \varphi represents latitude; \lambda represents longitude; and x and y are the projected (planar) coordinates for a given (\varphi, \lambda) coordinate pair. For example, the
sinusoidal projection is a very simple equal-area projection. Its generating formulae are: :\begin{align} x &= R \cdot \lambda \cos \varphi \\ y &= R \cdot \varphi \end{align} where R is the radius of the globe. Computing the partial derivatives, :\frac{\partial x}{\partial \varphi} = -R \cdot \lambda \cdot \sin \varphi,\quad \frac{\partial x}{\partial \lambda} = R \cdot \cos \varphi,\quad \frac{\partial y}{\partial \varphi} = R,\quad \frac{\partial y}{\partial \lambda} = 0 and so :\frac{\partial y}{\partial \varphi} \cdot \frac{\partial x}{\partial \lambda} - \frac{\partial y}{\partial \lambda} \cdot \frac{\partial x}{\partial \varphi} = R \cdot R \cdot \cos \varphi - 0 \cdot (-R \cdot \lambda \cdot \sin \varphi) = R^2 \cdot \cos \varphi = s \cdot \cos \varphi with s taking the value of the constant R^2. For an equal-area map of the
ellipsoid, the corresponding differential condition that must be met is: == List of equal-area projections ==