Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where $\mathbf{x}=(x,y,z)$ in $ \mathbb{R}^3$. Now consider:$$\mathbf{r}(u,v,t)=(x(u,v,t),y(u,v,t),z(u,v,t)) $$ where $(u,v)\in D\subset\mathbb{R}^2 $ ($t$ denotes time). Let $\mathbf{v}(u,v,t)=\partial \mathbf{r}(u,v,t)/\partial t.$
Well, according to my book:$$\int_{\partial D}(\mathbf{F}\times\mathbf{v})\cdot \frac{\partial \mathbf{r}}{\partial u}\,du+(\mathbf{F}\times\mathbf{v})\cdot \frac{\partial \mathbf{r}}{\partial v}\,dv=\int_{\partial D}(\mathbf{F}\times\mathbf{v})\cdot d\mathbf{s}$$ (Here $\mathbf{F}$ means $\mathbf{F}(\mathbf{r},t)$). So it should be:$$d\mathbf{s}=\frac{\partial \mathbf{r}}{\partial u}du+\frac{\partial \mathbf{r}}{\partial v}dv$$ Could someone explain me why is the latter expression valid? Shouldn't it be $d\mathbf{s}=(du,dv)$?