= Cotangent coordinate transition
{title2=$b=(D_xy)^{-T}a$}
For coordinates $x,y$ on a <smooth manifold>, fibre coefficients of a <cotangent space> element obey $b_j=\sum_i a_i\partial x^i/\partial y^j$. Thus induced <cotangent bundle> charts change by $(x,a)\mapsto(y(x),b)$ with smooth, invertible, fibre-linear changes. The <canonical one-form on a cotangent bundle> satisfies $\sum_i a_i dx^i=\sum_j b_jdy^j$, and differentiating proves invariance of its <exterior derivative>.
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