When is a curve “straight”? When its tangent vector is constant. We have 1) \begin{equation} \vv = \frac{d\rr}{d\lambda} = \dot\rr \end{equation} or equivalently \begin{equation} \vv\,d\lambda = d\rr \end{equation} from which the components of $\vv$ can be determined.

To determine whether $\vv$ is constant, we need to be able to differentiate it, and to do that we need to know how to differentiate our basis vectors, which are only constant in rectangular coordinates. For elementary examples, such as those considered in this chapter, these derivatives can be computed by expressing each basis vector in rectangular coordinates, differentiating, and converting the answer back to the coordinate system being used. In general, however, this step requires additional information; see § Connections for further details.

Once we know how to differentiate $\vv$, we can define a geodesic to be a curve satisfying \begin{equation} \dot\vv = \zero \label{geoeq} \end{equation} Since $\vv$ is itself a derivative, the geodesic equation is a set of coupled second-order ordinary differential equations in the parameter $\lambda$.