[list][*]Fixed values of the parameters [i]u[/i] and [i]v[/i] determine a point Q on the surface.[/*][*]Fixing [i]u[/i] but not [i]v[/i] creates a parametric curve through Q that lies on the surface, and fixing [i]v[/i] but not [i]u[/i] creates another curve on the surface through Q.[/*][*]The partial derivatives [math]\vec{r}_u[/math] and [math]\vec{r}_v[/math] are vectors parallel to these curves, and thus also parallel to the surface. That means they define a tangent plane to the surface at Q.[/*][*]Choosing two small step sizes defines two additional curves that enclose a small "panel" on the surface.[/*][*]By using the step sizes to scale the tangent vectors, we can define a parallelogram in the tangent plane tangent to S at Q. So long as the step sizes are small, the area of that parallelogram closely approximates the area of the enclosed panel on the surface.[/*][*]We can calculate the area of this parallelogram using cross products: [/*][/list][center][math]A=\left|\left|\Delta u\vec{r}_u\times\Delta v\vec{r}_v\right|\right|=\left|\left|\vec{r}_u\times\vec{r}_v\right|\right|\Delta u\Delta v[/math][/center][list][*]Summing over all the "pansl" on the surface gives an estimate of the total surface ares:[/*][/list][center][math]S=\sum\sum\left|\left|\vec{r}_u\times\vec{r}_v\right|\right|\Delta u\Delta v[/math][/center][list][*]Taking the limit as Δu and Δv go to zero gives [math]S=\int\int_sdS=\int\int_S\left|\left|\vec{r}_u\times\vec{r}_v\right|\right|dA[/math][/*][/list]