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Mathematics-Online lexicon:

Surface Area of a Graph of a Function


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Let $ f = f(x,y)$ be a differentiable scalar function defined on a region $ D
\subset \mathbb{R}^2 .$ Then

$\displaystyle \sigma: D \longrightarrow \mathbb{R}^3 \ \ \ $   defined by$\displaystyle \ \ \ (x,y) \mapsto (x,y,f(x,y)) $

is a regular parametrization of its graph.

The normal vector of this parametrization is

$\displaystyle \sigma_x \times \sigma_y \ = \ (-f_x, -f_y, 1) .$

Therefore the surface area of the graph is given by the integral

$\displaystyle \iint_D \sqrt{1 + f_x^2 + f_y^2} \ dx dy .$

()

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  automatically generated 7/ 6/2005