16.2 Line Integrals (Scalar Functions) PDF

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16.2 Line Integrals (Scalar Functions) Thursday, March 25, 2021

6꞉01 PM

Line Integral of a Scalar Function -

Integrals can be taken along a curve C thru a scalar function (scalar field).  ∫" 𝑓 (𝑥, 𝑦 )𝑑𝑠 where ds means "w.r.t. arc length" 𝑑𝑠 = |𝑟-# (𝑡 )|𝑑𝑡  ○ ∫" 1𝑑𝑠 = arc length of C We only care about the values of f(x,y) that are over the curve C ○ The height of a curtain over C that is f(x,y) tall 

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○ So ∫" 𝑓𝑑𝑠 = ∫% 𝑓 0𝑟-(𝑡 )12 |𝑟-# (𝑡 )|2𝑑𝑡 The orientation of C does not matter; it is still the same "curtain-path".

Projections -

We can isolate one component or another of the curve in order to integrate its project

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