16.2. Line Integrals
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1 16. Line Integrals
2 Review of line integrals: Work integral Rules: Fdr F d r = Mdx Ndy Pdz FT r'( t) ds r t since d '(s) and hence d ds '( ) r T r r ds T = Fr '( t) dt since r r'( ) dr d dt t dt dt does not depend on the parametrization. 1 if F Mi Nj Pk M, N, P dr dxi dyj dzk If consists of two paths and, then Fdr Fdr Fdr 1 If is traversed in the opposite direction, then Fdr Fdr
3 Outward Flux Let be a simple closed curve in the plane: r( t) x( t), y( t) (simple means it does not cross itself) Traverse counter clockwise. T is the (unit) tangent vector, and n normal to T given by n Tk n is the "outward pointing" normal a unit vector in the plane If T x '( t ), y '( t ) '( ), x'( ) then n y t t r'( t) r'( t) Definition : The outward flux across is F nds if My ' Nx ' F Mi Nj then F nds ds r '( t) My ' Nx ' r '( t ) dt r '( t) My ' Nx ' dt flux integral F nds My ' Nx ' dt if F Mi Nj whereas work integral is FT ds Mx ' Ny ' dt
4 outward flux: Fn ds My ' Nx ' dt A point P is called a source (or sink ) if Fn ds 0 (or < 0) for all sufficiently small circles centered at P.
5 Example : for the two force fields Find the outward flux F nds across the ellipse 4x 9y 1 F xi yj and F yi xj parametrize the ellipes: x cos( t), y sin( t), 0 t for 1 we get: My ' Nx ' xy ' yx ' cos( t) cos( t) sin( t) F sin( t) F1 nds dt positive outward flux F nds My ' Nx ' dt if F Mi Nj cos ( t) sin ( t) for we get: My ' Nx ' yy ' xx ' sin( t) cos( t) cos( t) F sin( t) sin( t)cos( t) sin( t)cos( t) F sin( )cos( ) sin ( ) nds t t dt t zero outward flux
6 16.3 Path Independence onservative vector fields
7 Definition : A vector field F is called conservative, if there exists a fuction f, such that F f. The function f is called the potential of F. A conservative field F is also called a gradient field. If F f, then F dr f f f dx dy dz x y z i j k i j k f f f dx dy dz x y z f dx f dy f dz dt x dt y dt z dt df dt f ( B ) f ( A ) dt
8 Definition : If F is a vector field, and a path from A to B, then F dr is called path independent if its value does not depend on the path from A to B. F dr = 1 F dr Fundamental theorem for line integrals : a) F f if and only if F dr is path independent. b) If F f, then F dr f ( B) f ( A) If so, we somtimes denote F dr = F dr if is a path from A to B. B A
9 Notice that F f f y i x j is conservative with potential f ( x, y) xy : f y x x i y j i j F xy (1,1) Hence dr ydx xdy f dr f ( B) f ( A) followed by a line from 1,0 to 1,1 Example : Evaluate ydx xdy, where is a line segments from 0,0 to 1,0 ydx xdy ydx xdy ydx xdy x t y 0 : 0 t 1 dx dt dy 0 1 x 1 y t : 0 t 1 dx 0 dy dt ydx xdy 0 0 dt 0, ydx xdy 0 1 dt 1, 0 1 ydx xdy (0,0) an also choose any path from (0, 0) to (1,1), e.g. : x t y t t dx dt dy dt ydx xdy 3 0 tdt tdt 1 0 tdt 1
10 Question : When is a vector field F conservative? Need: F = f f M i N j f x i y j or: f M x and N f y notice then: M f N f and y yx x xy M a necessary condition: y N x In 3 dimensions: f f f F = Mi Nj Pk f i j k x y z f f f need: M, N and P x y z M N M P N P necessary conditions:,, y x z x z y Question : Are these conditions sufficient?
11 Some terminology for curves : Definition : A region R is called simply connected can be shrunk to a point by curves staying in R. if every closed curve in R Important examples : The plane minus the origin is not simply connected. 3-space minus the origin is simply connected. 3-space minus the z-axis is not simply connected.
12 Theorem: If F = Mi Nj is defined in a connected and simply connected region, then F M f if and only if y N x Theorem: If F = Mi Nj Pk is defined in a connected and simply connected region, then F f if and only if M N M P N P,, y x z x z y Hint: To find the potential f, integrate one at a time f f f M, N (and P if in 3-space) x y z Definition : A differential Mdx Ndy Pdz is called exact if Mdx Ndy Pdz df for some function f. This is equivalent to saying that Mi Nj Pk f (can use same theorems)
13 Example : Find the work done by the force y y x y x e y xe F, i 4 j along the indicated curve. y M x e N 4y xe y y y M e N e so F y x is a gradient field. need: f x M and f y N, y y f x y Mdx x e dx x xe G y y, y, f x y xe G y N x y 4 y y xe G y y xe 4 G y y G y y y x xe y,0,,0 Work F dr f x y, ,0, y f x y x xe y
14 Example : Determine wether the given vector field is a gradient field. If so, find a potential function. xy 3xz x y y 3x z z F i j j F f if f M, f N, and f P x y z and this is possible if and only if M N, M P, and N P M N x y 4xy M z 6xz N z 0 4xy P 6xz P 0 x f Mdx xy 3xz dx f Ndy x y y dy f Pdz 3x z z dz 3,, y y x z x z y So F is a gradient field, Ff Now let's find f. x y x z G y, z 3 x y y H x, z 3 K x, y x z z f x y z x y x z y z, M xy 3xz N x y y P 3x z z G y z y z 3 H x, z x z z K x, y x y y
15 Note : To indicate that the line integral is over a closed curve, Theorem: F is conservative if and only if for every closed curve : F dr 0 Indeed, if F f and goes from A to B and A B then F dr f ( B) f ( A) 0 we often write 1 Fdr F dr onversely, assume F dr 0 for any closed curve and let and be two curves from A to B with A B Then 0 F dr F dr F dr and hence 1 1 Fdr Fdr 1
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