Proof of Vector Identity fro a Vector Field on a Surface Which is a Function of Time and a Position Also a Function of Time
For a vector field
\[\mathbf{F}(\mathbf{r} , t)\]
where \[\mathbf{r} = r(u,v,t)\]
\[\frac{\partial \mathbf{F}}{\partial v} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial u}) -\frac{\partial \mathbf{F}}{\partial u} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial v}) =((\mathbf{\nabla} \cdot \mathbf{F} ) \mathbf{v}- (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{F}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial u}) \]
where
\[\mathbf{v} = \frac{\partial \mathbf{r}}{\partial t}\]
.Proof
From the Chain Rule
\[\frac{\partial \mathbf{F}}{\partial v} =(\frac{\partial \mathbf{r}}{\partial v} \cdot \mathbf{\nabla}) \mathbf{F}, \: \frac{\partial \mathbf{F}}{\partial u} =(\frac{\partial \mathbf{r}}{\partial u} \cdot \mathbf{\nabla}) \mathbf{F} \]
\[\begin{equation} \begin{aligned} \frac{\partial \mathbf{F}}{\partial v} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial u}) -\frac{\partial \mathbf{F}}{\partial u} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial v}) &= \frac{\partial \mathbf{F}_i}{\partial v} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial u})_i -\frac{\partial \mathbf{F}_i}{\partial u} \cdot (\mathbf{\nabla} \times \frac{\mathbf{r}}{\partial v})_i \\ &= e_{ijk} \frac{\partial \mathbf{F}_i}{\partial v}v_j \frac{\partial x_k}{\partial u} - e_{ijk} \frac{\partial \mathbf{F}_i}{\partial u}v_j \frac{\partial x_k}{\partial v} \\ &= e_{ijk} \frac{\partial x_l}{\partial v}\frac{\partial \mathbf{F_i}_i}{\partial x_l}v_j \frac{\partial x_k}{\partial u} - e_{ijk} \frac{\partial x_l}{\partial u} \frac{\partial \mathbf{F}_i}{\partial x_l}v_j \frac{\partial x_k}{\partial v} \\ &= e_{ijk}v_j \frac{\partial \mathbf{F_i}_i}{\partial x_l} (\frac{\partial x_l}{\partial v} \frac{\partial x_k}{\partial u} - \frac{\partial x_l}{\partial u} \frac{\partial x_k}{\partial v}) \\ &= e_{ijk}v_j \frac{\partial \mathbf{F_i}_i}{\partial x_l} (\delta_{lm} \delta_{kn} - \delta_{ln} \delta_{km} ) \frac{\partial x_m}{\partial v}\frac{\partial x_n}{\partial u} \\ &= e_{ijk}v_j \frac{\partial \mathbf{F_i}_i}{\partial x_l} e_{lkr} e_{mnr} \frac{\partial x_m}{\partial v}\frac{\partial x_n}{\partial u} \\ &= e_{ijk}e_{rlk} e_{mnr} v_j \frac{\partial \mathbf{F_i}_i}{\partial x_l} \frac{\partial x_m}{\partial v}\frac{\partial x_n}{\partial u} \\ &= (\delta_{ir} \delta_{jl} - \delta_{il} \delta_{jr}) e_{mnr} v_j \frac{\partial \mathbf{F_i}_i}{\partial x_l} \frac{\partial x_m}{\partial v}\frac{\partial x_n}{\partial u} \\ &= e_{mnr} \frac{\partial x_m}{\partial v} \frac{\partial x_n}{\partial u} v_l \frac{\partial F_r}{\partial x_l} - e_{mnj} \frac{\partial x_m}{\partial v} \frac{\partial x_n}{\partial u} v_j \frac{\partial F_l}{\partial x_l} \\ &= -e_{rnm} v_l \frac{\partial F_r}{\partial x_n} \frac{\partial x_m}{\partial v} \frac{\partial x_n}{\partial u} +e_{jnm} v_j \frac{\partial F_l}{\partial x_l} \frac{\partial x_m}{\partial v} \frac{\partial x_n}{\partial u} \\ &= -(( \mathbf{v} \cdot \mathbf{\nabla} ) \mathbf{F})_r (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})_r + (( \mathbf{\nabla} \cdot \mathbf{F} ) \mathbf{v})_j (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})_j \\ &= ((\mathbf{\nabla} \cdot \mathbf{F} ) \mathbf{v}- (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{F}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial u}) \end{aligned} \end{equation}\]
