Line Integrals (4A Line Integral Path Independence
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Line Integral In the Plane x = f (t d x d t = f '(t d x = f '(t y = g(t d y = g'(t d y = g'(t d s = [ f '(t] 2 + [g'(t] 2 Curve C a t b d s d y d x G(x, y d x = a b G(f (t, g(t f '(t G(x, y d y = a b G(f (t, g(t g'(t G(x, y d s = a b G(f (t, g(t [ f '(t] 2 + [g'(t] 2 Line Integral 3
Line Integral In Space x = f (t d x d t = f '(t d x = f '(t y = g(t d y = g'(t d y = g'(t z = h(t d z = h'(t d z = h'(t Curve C a t b d s = [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 d h G(x, y, z d z = a b G(f (t, g(t, h(t h'(t d f d t d g G(x, y, z d s = a b G(f (t, g(t, h(t [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 Line Integral 4
Line Integral using r(t Arc Length Parameter s increases in the direction of increasing t t s(t = v (τ d τ t 0 t = t 0 r '(τ d τ t = t 0 [ f '(τ] 2 + [g'(τ] 2 + [h'(τ] 2 d τ d s = v(t d s = [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 G(x, y, z d s = a b G( r (t r '(t = a b G(f (t, g(t, h(t v (t = a b G(f (t, g(t, h(t [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 Line Integral 5
Line Integral with an Explicit Curve Function y = f (x a x b Curve C d y d x = f '(x d y = f '(x d x d s = [d x] 2 + [d y ] 2 d s = 1 + [ f '(x] 2 d x d s d x d y G(x, y d x = a b G(x, f (x d x G(x, y d y = a b G(x, f (x f '(x d x G(x, y d s = a b G(x, f (x 1 + [ f '(x] 2 d x Line Integral 6
Line Integral Notation In many applications G(x, y d s = P(x, y d x + Q(x, y d y = P(x, y d x + Q(x, y d y = P d x + Q d y G(x, y, z d s = P(x, y, z d x + Q(x, y, z d y + R(x, y, z d z Line Integral 7
Line Integral over a 2-D Vector Field (1 a given point in a 2-d space (x 0, y 0 A vector P (x 0, y 0, Q(x 0, y 0 domain y y range 2 functions (x 0, y 0 P (x 0, y 0 (x 0, y 0 F (x 0, y 0 x x (x 0, y 0 Q(x 0, y 0 (x 0, y 0 only points that are on the curve r (t = f (ti + g(t j x = f (t y = g(t a t b F (x 0, y 0 = P (x 0, y 0 i + Q(x 0, y 0 j Line Integral 8
Line Integral over a 2-D Vector Field (2 r (t = f (t i + g(t j dr d x = f '(ti + g'(t j = i + d y j dr = ( d x i + d y j = d x i + d y j dr = d x i + d y j F (x, y = P(x, y i + Q(x, y j F dr = P(x, y d x + Q(x, y d y c F dr = P(x, y d x + Q(x, y d y Line Integral 9
Line Integral over a 3-D Vector Field (1 A given point in a 3-d space (x 0, y 0, z 0 A vector P (x 0, y 0, z 0, Q( x 0, y 0, z 0, R(x 0, y 0, z 0 domain range 3 functions z (x 0, y 0, z 0 z (x 0, y 0, z 0 P (x 0, y 0, Z 0 F (x 0, y 0, z 0 (x 0, y 0, z 0 Q(x 0, y 0, z 0 (x 0, y 0, z 0 R(x 0, y 0, z 0 y y x x (x 0, y 0, z 0 only points that are on the curve r (t = f (ti + g(t j + h(t j x = f (t y = g(t z = h(t a t b F (x 0, y 0, z 0 = P (x 0, y 0, z 0 i + Q(x 0, y 0, z 0 j + R(x 0, y 0, z 0 k Line Integral 10
Line Integral over a 3-D Vector Field (2 r (t = f (t i + g(t j + h(t k dr d x = f '(ti + g'(t j + h'(t k = i + d y j + d z k dr = ( d x i + d y j + d z k = d x i + d y j + d z k dr = d x i + d y j + d z k F (x, y, z = P(x, y, z i + Q(x, y, z j + R(x, y, z k F dr = P(x, y, z d x + Q(x, y, z d y + R(x, y, z d z c F dr = P(x, y, z d x + Q(x, y, z d y + R(x, y, z d y Line Integral 11
Line Integral in Vector Fields r (t = f (t i + g(t j dr = d x i + d y j F (x, y, z = P i + Q j r (t = f (t i + g(t j + h(t k dr = d x i + d y j + d z k F (x, y, z = P i + Q j + R k c F dr = P d x + Q d y c F dr = P d x + Q d y + R d z P (x, y P (x, y, z Q(x, y Q(x, y, z R(x, y, z Line Integral 12
Work (1 W = F d A force field F (x, y = P (x, yi + Q(x, y j A smooth curve C :x = f (t, y = g(t, a t b Work done by F along C W = c F (x, y dr = P(x, y d x + Q(x, y d y dr = dr d s d s dr = d r d s d s dr = T d s W = c F dr = c F T d s Line Integral 13
Work (2 dr = dr d s d s dr = d r d s d s dr = T d s W = c F dr t 0 dr = t F 1 t 1 = t 0 = c F T d s ( P d f + Q d g + R dh F (x, y, z = P i + Q j + Rk = P(x, y, zi + Q(x, y, z j + R(x, y, zk t 1 = t 0 t 1 = t 0 ( P d x d t + Q d y + R d z d t P d x + Q d y + R d z r (t = f (ti + g(t j + h(tk x = f (t y = g(t z = h(t Line Integral 14
Circulation A Simple Closed Curve C Circulation circulation = C F dr = C F T d s Assume F is a velocity field of a fluid Circulation : a measure of the amount by which the fluid tends to turn the curve C by rotating around it C C C F dr = C F T d s = 0 C F dr = C F T d s > 0 Line Integral 15
Path Independence C1 C2 1 F dr 2 F dr In general 1 F dr = 2 F dr Exceptional Case Path Independence for a special kind of a vector field F Conservative Vector Field If we can find a scalar function Φ that satisfies F = Φ F is a gradient field of a scalar function Φ Line Integral 16
Conservative Vector Field F can be written as the gradient of a scalar function Φ F = Φ A vector function F in 2-d or 3-d space is conservative F (x, y = P (x, y i + Q(x, y j F (x, y, z = P (x, y, zi + Q(x, y, z j + R(x, y, z k If Φ(x, y satisfies If Φ(x, y, z satisfies Φ( x, y = Φ x i + Φ y j = P (x, y i + Q(x, y j Φ(x, y, z = Φ x i + Φ y j + Φ z k = P (x, y, zi + Q(x, y, z j + R(x, y, zk Φ(x, y = P (x, y x Φ (x, y = Q (x, y y Φ(x, y, z = P (x, y, z x Φ (x, y, z = Q(x, y, z y Φ(x, y, z = R(x, y, z z Line Integral 17
Fundamental Line Integral Theorem (1 F can be written as the gradient of a scalar function Φ F = Φ A vector function F in 2-d or 3-d space is conservative Conservative vector field F (x, y = P(x, y i + Q(x, y j Φ dr = Φ(B Φ( A A = ( f (a,g(a, B = ( f (b,g(b a b f '(x d x = f (b f (a F = Φ think as a differentiation Φ (x, y F dr = Φ dr = Φ(B Φ( A Line Integral 18
Fundamental Line Integral Theorem (2 Conservative vector field F (x, y = P(x, y i + Q(x, y j F dr = Φ dr = Φ(B Φ( A A = ( f (a,g(a B = ( f (b,g(b F ( x, y = Φ( x, y = Φ x i + Φ y j F dr = F dr (t = F r '(t = ( Φ x i + Φ y j ( d x i + d y j = ( Φ x d x + Φ y d y = ( Φ = [Φ (x(t, y (t] a b = Φ( x(b, y (b Φ(x (a, y (a = Φ( B Φ( A Line Integral 19
Connected Region (1 Connected Every pair of points A and B in the region can be joined by a piecewise smooth curve that lies entirely in the region Simply Connected Connected and every simple closed curve lying entirely within the region can be shrunk, or contracted, to a point without leaving the region The interior of the curve lies also entirely in the region No holes in the region Disconnected Cannot be joined by a piecewise smooth curve that lies entirely in the region Multiply Connected Open Connected Many holes within the region Contains no boundary points Line Integral 20
Connected Region (2 Connected Disconnected Simply Connected Multiply Connected Open Connected Line Integral 21
Equivalence In an open connected region Path Independence F d r Conservative Closed path C F = Φ C F dr = 0 Line Integral 22
Test for a Conservative Field F (x, y = P (x, y i + Q(x, y j P y = Q x : conservative vector field in an open region R F (x, y = P (x, y i + Q(x, y j : conservative vector field in R P y = Q x for all points in a simply connected region R F = Φ = Φ x i + Φ y j P = Φ x Q = Φ y P y = Q x P y = 2 Φ x y Q x = 2 Φ x y Line Integral 23
Equivalence in 3-D In an open connected region Path Independence F d r Conservative Closed path C F = Φ C F dr = 0 P y = Q x P z = R x Q z = R y curl F = ( P y Q x i + ( P z R x j + ( Q z R y k F = P i + Q j + R k F = Φ = Φ x i + Φ y j + Φ z k Line Integral 24
2-Divergence Flux across rectangle boundary ( M x Δ x Δ y + ( N y Δ y Δ x = ( M x + N y Δ x Δ y Flux density = ( M x + N y Divergence of F Flux Density Line Integral 25
References [1] http://en.wikipedia.org/ [2] http://planetmath.org/ [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] D.G. Zill, Advanced Engineering Mathematics