In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ are $\Delta u_i r_u$ and $\Delta v_i r_v$?
2025-01-12 23:32:36.1736724756
Area of parametric surface (theory)
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They aren't. Moreover, $\Delta T_i$ need not be a parallelogram at all, nor should be expect it to be any sort of planar region.
The region $\Delta T_i$ is approximated by the parallelogram you describe. The idea here is that as we let $\Delta u_i, \Delta v_j$ go to zero, the difference between the area of this parallelogram and the area of $\Delta T_i$ becomes negligible.