{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec7-1 - F Β΄ Β Λ† k = βˆ‚ F y βˆ‚ x-βˆ‚ F x βˆ‚ y β€’ So...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Green’s theorem in the plane We have seen how to do double integrals, now let’s do them in the xy-plane over an area A Z b a dx Z y u y l P ( x , y ) y dy = Z b a [ P ( x , y u ) - P ( x , y l )] dx = - I C Pdx The integral H C Pdx means a counterclockwise integral around the curve that bounds area A We could also do the same thing but integrate first with respect to y Chapter7: Fourier series
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Continued, Green’s theorem in the plane Z d c dy Z x r x l Q ( x , y ) x dx = Z d c [ Q ( x r , y ) - Q ( x l , y )] dy = - I C Qdy Putting these together we get Green’s theorem in the plane, Z Z A Q x - P y dxdy = I A ( Pdx + Qdy ) Chapter7: Fourier series
Background image of page 2
Green’s theorem in the plane, connection to line integrals and work Z Z A Q x - P y dxdy = I A ( Pdx + Qdy ) If we take F x = P and F y = Q , then we see Green’s theorem in the plane gives Z Z A F y x - F x y dxdy = I A ( F x dx + F y dy ) = W Then notice that ∇ ×
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: F Β΄ Β· Λ† k = βˆ‚ F y βˆ‚ x-βˆ‚ F x βˆ‚ y β€’ So if βˆ‡ ~ F = 0, then W = 0 as we already know Chapter7: Fourier series Divergence theorem in two dimensions β€’ Consider a vector field ~ V = V x Λ† i + V y Λ† j (Notice here V z = 0) β€’ If we take Q = V x and P =-V y , then βˆ‚ Q βˆ‚ x-βˆ‚ P βˆ‚ y = βˆ‚ V x βˆ‚ x + βˆ‚ V y βˆ‚ y = div ~ V β€’ Consider the outward normal Λ† n = Λ† idx-Λ† jdy √ dx 2 + dy 2 = Λ† idx-Λ† jdy ds , and then Pdx + Qdy =-V y dx + V x dy = ( V x Λ† i + V y Λ† j ) Β· ( Λ† idy-Λ† jdx ) = ~ V Β· Λ† nds β€’ The by Green’s theorem in the plane, Z Z A div ~ V dxdy = Z βˆ‚ A ~ V Β· Λ† nds Chapter7: Fourier series...
View Full Document

{[ snackBarMessage ]}