Consider two different charts \((U,\varphi)\) and \((V,\psi)\) on a manifold \(M\) with \(p\in U\cap V \neq \emptyset\). Assuming \((x^i)\) and \((\widetilde{x}^i)\) are the corresponding local coordinates, we obtain on \(T_pM\)

\begin{equation*} \frac{\partial }{\partial x^i}\bigg|_p=\frac{\partial \widetilde{x}^{j}}{\partial x^i}(p)\frac{\partial }{\partial \widetilde{x}^j}\bigg|_p, \end{equation*}

where \(\widetilde{x}^j=\psi\circ \varphi^{-1}\) in the numerator.

This implies that the components of a vector

\begin{equation*} v=v^i\frac{\partial }{\partial x^i}|_p \end{equation*}

are given by

\begin{equation*} \widetilde{v}^j=\frac{\partial \widetilde{x}^j}{\partial x^i}(p) v^i. \end{equation*}

Since the components of \(v\) transforms in the opposite way as the coordinate partial derivatives, tangent vectors are called contravariant vectors.

References Link to heading

1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.