Mathematics-Online lexicon: Flux through a Graph of a Function

	[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematics-Online lexicon:
	Flux through 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

overview

The flux into the direction of the

-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

of a differentiable scalar function

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 denotes a unit normal into the direction of $\sigma_x \times \sigma_y .$

()

Examples:

Beispiel: Fluss durch einen Funktionsgraph (german)
Fluss eines konstanten Vektorfeldes durch eine Ebene (german)

[Annotations] [Links]

automatically generated 7/ 6/2005