Let [math]\mathbf{F}[/math] be a vector field in three-dimensional space with continuous components defined over a smooth surface [math]S[/math] having a chosen field of normal unit vectors [math]\mathbf{n}[/math] orienting [math]S[/math]. Then [b]the surface integral of [/b][math]\mathbf{F}[/math][b] over [/b][math]S[/math] is [br] [math]\iint_{S} \mathbf{F} \cdot \mathbf{n}\ d\sigma[/math][br][br]This integral is also called the [b]flux [/b]of the vector field [math]\mathbf{F}[/math] across [math]S[/math]. [br][br]This interactive figure calculates flux for parametrized surfaces. Be sure to click both the [b]Surface [/b]button or the [b]Field [/b]button to define your surface parametrization and your vector field definition.

[i]This applet was developed for use with [url=https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable/P200000009666]Interactive Calculus[/url], published by Pearson.[/i]