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| [[Calculus/ParametricEquation|Parameterize]] ''f'' using ''r(u,v)'' for ''a ≤ u ≤ b'', ''c ≤ v ≤ d'' to get: | [[Calculus/ParametricEquation|Parameterize]] ''f'' using ''r(u,v)'' to get: |
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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]''. |
Surface Integral
A surface integral is an integral along a smooth curve C.
Contents
Scalar Surface Integral
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.
Parameterize f using r(u,v) to get:
where r = [x(u,v) y(u,v) z(u,v)], ru = [∂x/∂u ∂y/∂u ∂z/∂u], and rv = [∂x/∂v ∂y/∂v ∂z/∂v].
This gives a straightforward calculation for surface area:
