Exercise 2.05

Give an example of two linearly independent, nowhere-vanishing vector fields in $\mathbb{R}^{2}$ whose commutator does not vanish. Notice that these fields provide a basis for the tangent space at each point, but it cannot be a coordinate basis since the commutator doesn't vanish.

If we choose the following two linearly independent, nowhere-vanishing vector fields, $$ X^{i}=\begin{pmatrix} -x^{1} \\ 1 \end{pmatrix} \qquad \qquad Y^{i}=\begin{pmatrix} 1 \\ x^{1} \end{pmatrix} $$ the commutator does not vanish. $$ \begin{align} [X,Y]^{i}&= X^{j}\partial_{j}Y^{i}-Y^{j}\partial_{j}X^{i} \\ \\ &= \begin{pmatrix} \left( -x^{1}\,\partial_{1} +1\,\partial_{2} \right)1 - \left( 1\,\partial_{1} +x^{1}\,\partial_{2} \right)(-x^{1}) \\ \left( -x^{1}\,\partial_{1} +1\,\partial_{2} \right)x^{1} - \left( 1\,\partial_{1} +x\,\partial_{2} \right)1 \end{pmatrix} \\ \\ &=\begin{pmatrix} 1 \\ -x^{1} \end{pmatrix} \end{align} $$ $X$ and $Y$ are orthogonal over $\mathbb{R}^{2}$, so they could provide a basis for the tangent space.