Mathematics-Online lexicon: Surface Area of a Graph of a Function

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	Mathematics-Online lexicon:
	Surface Area of a Graph of a Function

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Let

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

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

()

Examples:

automatically generated 7/ 6/2005