Let \(M^n\) be a manifold which is assigned two coordinate charts \((U)\) and \((V)\). These two coordinate charts have a non-empty overlap, so that any point \(p\) in their intersection \(U\cap V\) has two numerical representations. The coordinate transformation from \((V)\) to \((U)\) is characterized by the transition map \(\Phi _{UV}\). Also note that \((U)\) and \((V)\) may have different number of coordinate components. For example, for a unit sphere embedded in the 3D space \(\mathbb {R}^3\), we can use either Cartesian coordinates \((x,y,z)\) or altitude-azimuth angles \((\theta , \phi )\) to characterize it.

  1. The \(i\)-th column vector of the Jacobian matrix \(J_{UV}\) for the transition map \(\Phi _{UV}\), if evaluated at a point \(p\) in the overlap \(U\cap V\), is the \(i\)-th basis vector of the tangent space rooted at \(p\) in the chart \((V)\), which is numerically represented by the basis of the chart \((U)\). N.B. This basis vector in the tangent space may not be a normalized vector.

  2. If the coordinate chart \((V)\) has \(k\) components, there are \(k\) columns in the Jacobian matrix, which span an oriented parallelogram.

  3. For a fixed tangent vector at \(p\), the Jacobian matrix \(J_{UV}\) can transform its coordinates in \((V)\) to \((U)\).

    Coordinate transformation in robotics is just a special case, where both \((U)\) and \((V)\) are flat frames. If all frames have a common origin, i.e. there is only rotation but no translation between them, the transition map \(\Phi _{UV}\) is a rotation matrix, which is also the Jacobian matrix \(J_{UV}\). The rotation matrix from chart \((U)\) to \((V)\) can be constructed by representing each basis vector of chart \((V)\) in chart \((U)\). Each numerical representation of the basis vector is a column vector and all such column vectors are placed horizontally as a rotation matrix \(R\). This rotation matrix \(R\) is just the Jacobian matrix \(J_{UV}\) from \((V)\) to \((U)\) 1. Then for a fixed poitn \(p\), its coordinates in the two charts are related as \(P^U = RP^V\), which is consistent with \(P^U = J_{UV} P^{V}\).

  4. The Gramian matrix \(J^{\mathrm {T}} J\) is a metric tensor, whose entries are the inner products of pairs of the basis vectors of the tangent space, i.e. \begin{equation} (J^{\mathrm {T}}J)_{ij} = \langle \boldsymbol {\partial }x_i^V, \boldsymbol {\partial }x_j^V \rangle . \end{equation}

1N.B. Different conventions are used in robotics and differential geometry. In the former, the rotation matrix is from chart \((U)\) to \((V)\), while in the latter, the Jacobian matrix is from \((V)\) to \((U)\). They are actually a same thing.