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Mathematics-Online lexicon:

Flux through a Graph of a Function

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The flux into the direction of the $ z$-axis of a continuous vector field $ \Phi (x,y,z) = (p(x,y,z),q(x,y,z),r(x,y,z)) $ through the graph $ S$ of a differentiable scalar function $ f=f(x,y)$ defined on a region $ D\subseteq \mathbb{R}^2$ is given by the integral

$\displaystyle \iint\limits_{S} \Phi \cdot n d\sigma = \iint\limits_D \Phi(\sig... ...a_x \times \sigma_y) \ dx dy = \iint\limits_D -p f_x - q f_y + r \, dx dy \ . $

Here $ \sigma $ is the parametrization $ (x,y) \mapsto (x,y,f(x,y)) $ and $ n$ denotes a unit normal into the direction of $ \sigma_x \times \sigma_y .$

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  automatically generated 7/ 6/2005