Line Integrals (4A) Line Integral Path Independence. Young Won Lim 11/2/12
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1 Line Integrals (4A Line Integral Path Independence
2 Copyright (c 2012 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version pulished y the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to youngwlim@hotmail.com. This document was produced y using OpenOffice and Octave.
3 Line Integral of a Scalar Field 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 dr d t G(x, y, z d s r (t 0 r (t 0 +Δ t Δ r Δ s = a G( r (t r '(t = a G(f (t, g(t, h(t v (t = a G(f (t, g(t, h(t [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 Line Integral 3
4 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 Parameterized Curve C a t d s d x d y d s = [d x] 2 + [d y ] 2 d s = [ f '(t] 2 + [g'(t] 2 G(x, y d x = a G(f (t, g(t f '(t G(x, y d y = a G(f (t, g(t g'(t G(x, y d s = a G(f (t, g(t [ f '(t] 2 + [g'(t] 2 Line Integral 4
5 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 Parameterized Curve C d h d s = [d x] 2 + [d y ] 2 + [d z] 2 a t d f d t d g d s = [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 G(x, y, z d z = a G(f (t, g(t, h(t h'(t G(x, y, z d s = a G(f (t, g(t, h(t [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 Line Integral 5
6 Line Integral with an Explicit Curve Function y = f (x d y d x = f '(x d y = f '(x d x Explicit Curve Function a x d s d y d s = [d x] 2 + [d y ] 2 d x d s = 1 + [ f '(x] 2 d x G(x, y d x = a G(x, f (x d x G(x, y d y = a G(x, f (x f '(x d x G(x, y d s = a G(x, f (x 1 + [ f '(x] 2 d x Line Integral 6
7 Line Integral in the Plane (1 domain z R 2 G(r (t range G(f (t, g(t z R 3 G(x, f (t y y x x r (t G(x, y G( r (t x = f (t y = g(t G(x, y G(f (t, g(t y = f (x G(x, y G(x, f (x Line Integral 7
8 Line Integral in the Plane (2 domain z R 2 G(r (t range G(f (t, g(t z R 3 G(x, f (t y y x x r (t x = f (t y = g(t y = f (x G(x, y d s = a G( r (t r '(t G(x, y d s = a G(f (t, g(t [ f '(t] 2 + [g'(t] 2 G(x, y d s = a G(x, f (x 1 + [ f '(x] 2 d x Line Integral 8
9 Line Integral in the Space (1 domain z R 3 range R 4 G(r (t z G(f (t, g(t, h(t y y x x r (t (t G(x, y G( r (t x = f (t y = g(t z = h(t G(x, y G(f (t, g(t Line Integral 9
10 Line Integral in the Space (2 domain z R 3 range R 4 G(r (t z G(f (t, g(t, h(t y y x x r (t G(x, y, z d s = a G( r (t r '(t r (t x = f (t y = g(t z = h(t G(x, y, z d s = a G(f (t, g(t, h(t [ f '(t] 2 + [g'(t] 2 + [h'(t] 2 Line Integral 10
11 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 11
12 Line Integral of a Vector Field using r(t dr = v (t dr = [ f '(ti + g'(t j + h'(tk] F (x, y, z dr = a F ( r (t r '(t dr d t = a {P (r (ti + Q(r (t j + R(r (tk} r '(t r (t 0 +Δ t = a P(r (t f '(t + Q(r (tg'(t + R(r (th'(t r (t 0 Δ r Δ s = a P(r (t dx + Q(r (t dy + R(r (t dz = a P(r (tdx + Q(r (tdy + R(r (tdz Line Integral 12
13 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 F (x 0, y 0 = P (x 0, y 0 i + Q(x 0, y 0 j Line Integral 13
14 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 differentiate wrt t multiply dr = ( d x i + d y j = d x i + d y j dr = d x i + d y j inner product 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 14
15 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 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 15
16 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 dr = ( d x i + d y dr = d x i + d y j + d z k j + d z k = d x i + d y j + d z k j + d z k differentiate wrt t multiply inner product 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 16
17 Line Integral in Vector Fields r (t = f (t i + g(t j dr = d x i + d y j F (x, y = 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 17
18 Line Integral in 2-D Vector Fields domain z R 2 range R 2 z F ( x, y = P i + Q j y P (x, y Q(x, y y x x r (t Position Vector r (t = f (t i + g(t j dr = d x i + d y j F (x, y = P i + Q j a t c F dr = P d x + Q d y Line Integral 18
19 Line Integral in 3-D Vector Fields domain z R 3 range R 2 z F ( x, y, z = P i + Q j + R k x y P (x, y, z Q(x, y, z R(x, y, z x y r (t Position Vector 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 a t c F dr = P d x + Q d y + R d z Line Integral 19
20 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 Work done y 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 20
21 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 21
22 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 y which the fluid tends to turn the curve C y 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 22
23 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 23
24 Conservative Vector Field F can e 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 24
25 Fundamental Line Integral Theorem (1 F can e 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 (,g( a f '(x d x = f ( f (a F = Φ think as a differentiation Φ (x, y F dr = Φ dr = Φ(B Φ( A Line Integral 25
26 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 (,g( 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 = Φ( x(, y ( Φ(x (a, y (a = Φ( B Φ( A Line Integral 26
27 Connected Region (1 Connected Every pair of points A and B in the region can e joined y a piecewise smooth curve that lies entirely in the region Simply Connected Connected and every simple closed curve lying entirely within the region can e 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 e joined y a piecewise smooth curve that lies entirely in the region Multiply Connected Open Connected Many holes within the region Contains no oundary points Line Integral 27
28 Connected Region (2 Connected Disconnected Simply Connected Multiply Connected Open Connected Line Integral 28
29 Equivalence In an open connected region Path Independence F d r Conservative Closed path C F = Φ C F dr = 0 Line Integral 29
30 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 30
31 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 31
32 2-Divergence Flux across rectangle oundary ( 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 32
33 References [1] [2] [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] D.G. Zill, Advanced Engineering Mathematics
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