Let $\langle\,.\,,\,.\,\rangle$ be the standard metric on $\mathbb{C}\,$, and $g$ a positive real-valued function on $\mathbb{C}\,$. Then with respect to the metric $g\langle\,.\,,\,.\,\rangle$, the mean curvature vector of a curve $\gamma\subset\mathbb{C}\,$ is, in terms of the standard mean curvature vector $\MCV$ (and calculating the unit normal $\mathbf{n}$ in the standard metric),