Surface Area

Consider a surface in mathbf{R}^3 that is described by a function from the square C := [0,1] times [0,1] to mathbf{R}. The corresponding surface area is

 A(f) := int_C sqrt{1+|nabla f(x,y)|_2^2} : dxdy.

We can obtain this formula using a grid of the square C, with grid points (ih,jh), 0 le i,j le K, where h=1/K is a small increment. With this discretization, to a point on the grid, we associate a rectangle on the plane z = 0. The area of the rectangle is h^2, with h = 1/K the spacing of the grid. However, the area of the corresponding surface is not the same, as it is not a rectangle but a parallelogram.

This parallelogram is defined by its two adjacent vectors

 u = h left( begin{array}{c} 1  0  frac{partial f}{partial x}(ih,jh)  end{array} right), ;; v = h left( begin{array}{c} 0  1  frac{partial f}{partial y}(ih,jh)  end{array} right) .

The formula for the area of a parallelogram defined by two adjacent vectors u,v in mathbf{R}^3 is the magnitude of the cross-product between the two vectors:

 | u times v| = sqrt{|u|_2^2 cdot |v|_2^2 - (u^Tv)^2} = |u|_2 |v|_2 sin (theta),

where theta is the angle between the two vectors.

Using the expressions for u,v given above, after some algebra we obtain the formula for the area of the parallelogram:

 h^2 sqrt{1+frac{partial f}{partial x}(ih,jh)^2 + frac{partial f}{partial y}(ih,jh)^2} ,

which proves the formula above.