Double Integrals (5A)

Transcription

1 Doule Integrls (5A) Doule Integrl Doule Integrls in Polr oordintes Green's Theorem

2 opyright (c) 2012 Young W. Lim. Permission is grnted to copy, distriute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version pulished y the Free Softwre Foundtion; with no Invrint Sections, no Front-over Texts, nd no Bck-over Texts. A copy of the license is included in the section entitled "GNU Free Documenttion License". Plese send corrections (or suggestions) to youngwlim@hotmil.com. This document ws produced y using OpenOffice nd Octve.

3 Are nd Volume A = d A V = f (x, y)d A Vector lculus (5A) Doule Integrl 3

4 Type I nd Type II z z x g 1 y g 2 c y d h 1 (y) x h 2 ( y) y = g 1 y = g 2 y c d y x x x = h 1 (y) x = h 2 (y) Vector lculus (5A) Doule Integrl 4

5 Fuini's Theorem z z x g 1 y g 2 c y d h 1 (y) x h 2 ( y) y = g 1 y = g 2 c d y y x = h 1 (y) x f (x, y) da g 2 = g1 f (x, y) dy d x x f (x, y) da h 2 ( y ) = c d h1 ( y ) x = h 2 (y) f (x, y) dx d y Vector lculus (5A) Doule Integrl 5

6 Type A nd Type B θ = h 2 (r ) r = g 2 (θ) θ = h 1 (r ) β α r = g 1 (θ) r = r = f (r,θ) da g 2 (θ) = α β g1 (θ) f (r,θ) r dr dθ f (r,θ) da h 2 (r ) = h1 (r ) f (r,θ) dθ r dr Vector lculus (5A) Doule Integrl 6

7 Work using n Arc Length Prmeter s W = F d A force field F (x, y) = P (x, y)i + Q(x, y) j A smooth curve :x = f (t), y = g(t), t Work done y F long W = c F (x, y) dr = P(x, y) d x + Q(x, y) d y Unit Tngent Vector dr dt = dr d s d s dt dr = d r d s d s dr = T d s W = c F dr = c F T d s Vector lculus (5A) Doule Integrl 7

8 Green's Theorem in the Plne (1) : piecewise simple closed curve ounding y simply connected region P d x + Q d y = ( Q Line Integrl Doule Integrl y = g 2 d P y d y dx x = h 1 ( y) + + y = g 1 = P dx c Q x d x d y = d c Q x = h 2 ( y) d y y 2 x 2 f '(y)dy = f ( y 2 ) f (y 1 ) f 'dx = f (x 2 ) f (x 1 ) y 1 x 1 Vector lculus (5A) Doule Integrl 8

9 Line Integrl in the Plne (2) Line Integrl z P (x, y) z Q(x, y) P d x + Q d y y y x x Doule Integrl z P y (x, y) z Q x (x, y) ( Q y y x x Vector lculus (5A) Doule Integrl 9

10 Green's Theorem in the Plne (3) : piecewise c simple closed curve : simply connected ounding region P d x + Q d y = ( Q y = g 2 P d x d x = h 1 ( y) c x = h 2 ( y) y = g 1 P y d A Q d y Q x d A = = = = g 2 ( x) P g 1 ( x) y d y d x [ P(x,g 2 ) P (x,g 1 ( x)) ] d x P (x,g 1 ( x)) d x P d x P (x,g 2 ) d x Vector lculus (5A) Doule Integrl 10 d = c d = c d = c = h 2 ( y ) Q h 1 ( y ) x d x d y [ Q(h 2 ( y ), y) Q(h 1 ( y), y )] d y Q(h 1 ( y), y) d x Q d y P (h 2 ( y ), y) d x

11 egion with Holes : piecewise simple closed curve ounding y simply connected region P d x + Q d y = Line Integrl ( Q Doule Integrl ( Q = 1 ( Q ) y d A + ( Q 2 ) y d A 1 2 = 1 P d x + Q d y + 2 P d x + Q d y 1 1 = P d x + Q d y Vector lculus (5A) Doule Integrl 11

12 Vector Form of Green's Theorem url : piecewise simple closed curve ounding y simply connected region P d x + Q d y = Line Integrl ( Q Doule Integrl curl F = F = i j k x y z P Q 0 curl F = ( Q x P y ) k P d x + Q d y = ( F ) k d A ( F ) k = ( Q x P y ) Vector lculus (5A) Doule Integrl 12

13 Vector Form of Green's Theorem Div (1) : piecewise simple closed curve ounding y simply connected region P d x + Q d y = ( Q T Line Integrl n Doule Integrl ( F T ) d s ( F n) ds T n P d y Q d x d x ds d y d s T = d x d s i + d y d s j d x ds d y d s n = d y ds i d x d s j ( F T ) d s = ( F n) ds = P d x + Q d y P d y Q d x Vector lculus (5A) Doule Integrl 13

14 Vector Form of Green's Theorem Div (2) : piecewise simple closed curve ounding y simply connected region P d x + Q d y = ( Q T Line Integrl n Doule Integrl ( F T ) d s ( F n) ds T n P d y Q d x ( F T ) d s = P d x + Q d y = ( Q ( F n) ds = P d y Q d x = ( P x + Q Vector lculus (5A) Doule Integrl 14

15 Vector Form of Green's Theorem Div : piecewise simple closed curve ounding y simply connected region P d x + Q d y = ( Q n Line Integrl Doule Integrl P d y Q d x d x ds n d y d s d x ds d y d s div F = ( Q x P y ) k T = d x d s i + d y d s j n = d y ds i d x d s j Vector lculus (5A) ( F Doule Integrl 15 ) k = ( Q x P y )

16 2-Divergence Flux cross rectngle oundry ( 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 Vector lculus (5A) Doule Integrl 16

17 eferences [1] [2] [3] M.L. Bos, Mthemticl Methods in the Physicl Sciences [4] D.G. Zill, Advnced Engineering Mthemtics