Line Integrals dt k! ~Path Dependence~ ! “Work Done’ can be extremely different depending on the...
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Miss Millie Millie Line Integrals ~Arc Length~ ≈ !() ! + () ! + () ! ≈ ! ! ! ! + ! ! ! + ! ! ! ∴ = ! ! ! ! ! + ! ! ! + ! ! ! ! ! ~Work Done~ W = F ! ∙ d ! δW = F ! ∙ δr ! ! ! W = ! F ! (x(t), y(t), z(t)) ∙ dr ! ! dt dt !" !" W = ! F ! (x, y, z) ∙ dr ⬚ ! = ! F ! ∙ dr dt dt ⬚ ! remember… r !(t) = x(t)ı̂ + y(t)ȷ̂ + z(t)k ! ∴ dr ! dt = dx dt ı̂ + dy dt ȷ̂ + dz dt k ! ~Path Dependence~ “Work Done’ can be extremely different depending on the path taken. There are some fields, ! , for which ! ! ⋅ ̃ ⬚ ! = 0 for all closed loops. These fields are called conservative, and work done is always 0 ! = − ⏟
Transcript of Line Integrals dt k! ~Path Dependence~ ! “Work Done’ can be extremely different depending on the...
Miss Millie Millie
Line Integrals ~Arc Length~
𝛿𝑠 ≈ !(𝛿𝑥)! + (𝛿𝑦)! + (𝛿𝑧)!
≈ !!𝛿𝑥𝛿𝑡!!
+ !𝛿𝑦𝛿𝑡!!
+ !𝛿𝑧𝛿𝑡!!
𝛿𝑡
∴ 𝑠 = ! !!𝑑𝑥𝑑𝑡!!
+ !𝑑𝑦𝑑𝑡!!
+ !𝑑𝑧𝑑𝑡!!
𝑑𝑡!
!
~Work Done~
W = F!⃗ ∙ d!⃗
δW = F!⃗ ∙ δr!!!⃗
W = ! F!⃗ (x(t), y(t), z(t)) ∙dr !!⃗dtdt
!"
!"
W = ! F!⃗ (x, y, z) ∙ dr⃗⬚
!= ! F!⃗ ∙
dr⃗dtdt
⬚
!
remember…
r !(t) = x(t)ı ̂+ y(t)ȷ ̂+ z(t)k!
∴dr!dt=dxdtı ̂+
dydtȷ ̂+
dzdtk!
~Path Dependence~
“WorkDone’canbeextremelydifferentdependingonthepathtaken.Therearesomefields,𝐹! ,forwhich
! 𝐹! ⋅ 𝑑�̃�⬚
!= 0
forallclosedloops.Thesefieldsarecalledconservative,andworkdoneisalways0
𝐹! = − 𝛁⏟ 𝑽
![(Reference [2]) LINEAR PHASE LOCKED LOOPS - …users.ece.gatech.edu/.../ECE_6440/Summer_2003/L060-LPLL-II(2UP).pdf · (Reference [2]) LINEAR PHASE LOCKED LOOPS - CONTINUED THE ACQUISTION](https://static.fdocument.org/doc/165x107/5ad972fe7f8b9a52528b89b2/reference-2-linear-phase-locked-loops-usersece-2uppdfreference-2.jpg)
