We can parametrize a surface [math]S[/math] over a rectangular domain [math]R[/math]:[br] [math]\mathbf{r}\left(u,v\right)=\left\langle f\left(u,v\right),g\left(u,v\right),h\left(u,v\right)\right\rangle[/math] [math]a\le u\le b[/math] [math]c\le v\le d[/math].[br]Suppose that [math]\mathbf{F}[/math] is a vector field in three-dimensional space[br] [math]\mathbf{F}\left(x,y,z\right)=\left\langle A\left(x,y,z\right),B\left(x,y,z\right),C\left(x,y,z\right)\right\rangle[/math][br]defined at every point [math]\left(x,y,z\right)[/math] on surface [math]S[/math]. Let [math]\mathbf{n}[/math] the unit normal vectors normal to surface [math]S[/math]. [b]The surface integral of[/b] [math]\mathbf{F}[/math] [b]over the surface[/b] [math]S[/math] can be calculated[br] [math]\int\int_S\mathbf{F}\cdot\mathbf{n}d\sigma=\int\int_R\mathbf{F}\cdot\frac{\mathbf{r}_u\times\mathbf{r}_v}{\left|\mathbf{r}_u\times\mathbf{r}_v\right|}\left|\mathbf{r}_u\times\mathbf{r}_{_v}\right|dudv=\int\int_R\mathbf{F}\cdot\left(\mathbf{r}_u\times\mathbf{r}_v\right)dudv[/math].[br]This integral is called the flux of the vector field [math]\mathbf{F}[/math] across [math]S[/math].[br][br]In this interactive figure, move the smaller green point throughout the [math]uv[/math]-region [math]R[/math]. You will see the value of [math]\mathbf{F}\cdot\mathbf{n}[/math] at this point. [br][br]Move the larger blue point from bottom left to top right to "integrate." When the blue point is in the top-right corner, the value of the surface integral is shown (this is a numerical approximation).