Transformation of Coordinate Covectors with Respect to New Coordinates

\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Consider two different charts \((U,\varphi)\) and \((V,\psi)\) on a manifold \(M\) with \(p\in U\cap V \neq \emptyset\). Let \((x^i)\) and \((\widetilde{x}^i)\) are the corresponding local coordinates. Let \(\omega=\omega_idx^i\) be a covector. How the components transform with respect to new coordinates?

To answer this question we use the transformation rule for coordinate vectors given in (0x66b312dd) . We then obtain

\begin{equation*} \omega_i=\omega\Biggl(\frac{\partial }{\partial x^i}\bigg|_p\Biggr)=\omega\Biggl(\frac{\partial \widetilde{x}^j}{\partial x^i}(p)\frac{\partial }{\partial \widetilde{x}^j}\bigg|_p\Biggr)=\frac{\partial \widetilde{x}^j}{\partial x^i}(p)\widetilde{\omega}_j. \end{equation*}

Since the components of \(\omega\) transforms in the same way as the coordinate partial derivatives tangent covectors are called covariant vectors.