= Surface Integral =

A '''surface integral''' is an [[Calculus/Integral|integral]] along a smooth curve ''C''.

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== Description ==

Given a smooth surface ''S'', integrating a function ''f'' along ''S'' gives the '''scalar surface integral'''. As the name implies, this returns a scalar value.

{{attachment:scalar1.svg}}

[[Calculus/ParametricEquation|Parameterize]] ''f'' using ''r(u,v)'' to get:

{{attachment:scalar2.svg}}

where ''r = [x(u,v) y(u,v) z(u,v)]'', ''r,,u,, = [∂x/∂u ∂y/∂u ∂z/∂u]'', and ''r,,v,, = [∂x/∂v ∂y/∂v ∂z/∂v]''.

This gives a straightforward calculation for surface area:

{{attachment:sa.svg}}



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CategoryRicottone