Download - Line Integrals 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.

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Page 1: Line Integrals 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.

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

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