= Coordinate basis
{title2=$\partial/\partial x^\mu$}
= Coordinate frame
{synonym}
A <manifold chart> with coordinates $x^\mu$ induces a smooth <local frame> of the <tangent bundle>, with <tangent vectors> $\partial_\mu=\partial/\partial x^\mu$ forming a <basis> of each <tangent space>. Their <Lie brackets of vector fields> vanish because mixed <partial derivatives> of smooth functions commute. Under a change of coordinates they transform by the <chain rule>: $\partial/\partial y^\nu=(\partial x^\mu/\partial y^\nu)\partial/\partial x^\mu$, with coefficients given by the <Jacobian matrix>.
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