Divergence and Curl (3B)
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1 Divergence and Curl (3B Divergence Curl Green's Theorem
2 Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by 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 by using OpenOffice and Octave.
3 2-D Vector Field a given point in a 2-d space (x 0, y 0 A vector M (x 0, y 0, N (x 0, y 0 domain range 2 functions y y (x 0, y 0 M (x 0, y 0 (x 0, y 0 F (x 0, y 0 (x 0, y 0 N (x 0, y 0 x x (x 0, y 0 F (x 0, y 0 = M(x 0, y 0 i + N (x 0, y 0 j Divergence & Curl 3
4 3-D Vector Field A given point in a 3-d space (x 0, y 0, z 0 A vector M (x 0, y 0, z 0, N (x 0, y 0, z 0, P (x 0, y 0, z 0 domain range 3 functions z (x 0, y 0, z 0 z (x 0, y 0, z 0 M (x 0, y 0, Z 0 F (x 0, y 0, z 0 (x 0, y 0, z 0 N (x 0, y 0, z 0 (x 0, y 0, z 0 P (x 0, y 0, z 0 y y x x (x 0, y 0, z 0 F (x 0, y 0, z 0 = M(x 0, y 0, z 0 i + N (x 0, y 0, z 0 j + P (x 0, y 0, z 0 k Divergence & Curl 4
5 Inward & Outward Bound Divergence & Curl 5
6 2-D Divergence (5 Velocity Fields of fluid flows i j Left Velocity (x, y F (x, y = M (x, yi + N(x, y j Top Velocity (x, y+ F (x, y( j F (x, y(+ j = N(x, y F (x, y = +N (x, y F (x, y( i = M (x, y Right Velocity {M (x+, y M (x, y} = ( M x {N (x, y+ N (x, y} = ( N y (x+, y Bottom Velocity (x+, y+ F (x, y(+i = + M (x, y Flow rate of outward bound fluid Flux across rectangle boundary ( M x + ( N y = ( M x + N y Flux density = ( M x + N y Divergence of F Flux Density Divergence & Curl 6
7 2-D Divergence (d Flux across rectangle boundary ( M x + ( N y = ( M x + N y F (x, y Flux density = ( M x + N y Divergence of F Flux Density Divergence & Curl 7
8 Inward & Outward Bound Divergence & Curl 8
9 Inward & Outward Bound X (i F (x, y = M (x, yi + N (x, y j M (x, y {M (x+, y M (x, y} M (x +, y M (x, y M (x+, y M (x+, y M (x, y M x Divergence & Curl 9
10 Inward & Outward Bound X (ii F (x, y = M (x, yi + N (x, y j M (x, y {M (x+, y M (x, y} M (x +, y M (x, y M (x+, y M (x+, y M (x, y M x Divergence & Curl 10
11 Inward & Outward Bound X (iii F (x, y = M (x, yi + N (x, y j M (x, y M (x +, y {M (x+, y M (x, y} > 0 > 0 {M (x+, y M (x, y} < 0 > 0 Outward bound net flow Inward bound net flow M (x, y M(x +, y {M (x+, y M (x, y} M x > 0 Outward M(x +, y M (x, y M x {M (x+, y M (x, y} M x < 0 Inward Divergence & Curl 11
12 Inward & Outward Bound X (iv F (x, y = M (x, yi + N (x, y j M (x, y M (x +, y {M (x+, y M (x, y} M x > 0 Positive Slope of a tangent line parallel to the x axis Outward bound net flow along the x axis {M (x+, y M (x, y} M x < 0 Negative Slope of a tangent line parallel to the x axis Inward bound net flow along the x axis Divergence & Curl 12
13 Inward & Outward Bound Y (i F (x, y = M (x, yi + N (x, y j {M (x+, y M (x, y} N (x, y N (x, y + N (x, y N (x, y + N (x, y + N (x, y + N y Divergence & Curl 13
14 Inward & Outward Bound Y (ii F (x, y = M (x, yi + N (x, y j {M (x+, y M (x, y} N (x, y N (x, y + N (x, y N (x, y + N (x, y + N (x, y N y Divergence & Curl 14
15 Inward & Outward Bound Y (iii F (x, y = M (x, yi + N (x, y j N (x, y N (x, y + {N (x, y+ N (x, y } > 0 > 0 {N (x, y+ N (x, y } < 0 > 0 Outward bound net flow Inward bound net flow N (x, y N (x, y+ N (x, y + N (x, y 3 +3 N y {N (x, y + N (x, y} N y > 0 Outward +3 3 {N (x, y + N (x, y} N y < 0 Inward Divergence & Curl 15
16 Inward & Outward Bound Y (iv F (x, y = M (x, yi + N (x, y j N (x, y N (x, y + {N (x, y + N (x, y} N y > 0 Positive Slope of a tangent line parallel to the y axis Outward bound net flow along the y axis {N (x, y + N (x, y} N y < 0 Negative Slope of a tangent line parallel to the y axis Inward bound net flow along the y axis Divergence & Curl 16
17 Inward & Outward Bound Divergence & Curl 17
18 Inward & Outward Bound M (x, y N (x, y N (x, y + M (x +, y {M (x+, y M (x, y} M x > 0 Outward {M (x+, y M (x, y} M x < 0 Inward ( x i + y j ( M i {N (x, y + N (x, y} N y > 0 Outward {N (x, y + N (x, y} N y < 0 Inward ( x i + y j ( N j = (M i = (N j Divergence & Curl 18
19 2-D Divergence and Del Operator 2-D Vector Field F (x, y = M (x, yi + N (x, y j Flux across rectangle boundary ( M x + ( N y = ( M x + N y Flux density = ( M x + N y F (x, y = ( x i + y j (M i + N j = F Divergence of F Divergence & Curl 19
20 3-D Divergence and Del Operator 2-D Vector Field F (x, y = M (x, yi + N (x, y j Divergence of F = ( M x + N y = ( x i + y j (M i + N j = F 3-D Vector Field F (x, y, z = M(x, y, zi + N (x, y, z j + P (x, y, zk Divergence of F = ( M x + N y + P z = ( x i + y j + z k (M i + N j + P k = F Divergence & Curl 20
21 Divergence & Curl 21
22 2-D Curl (4 Velocity Fields of fluid flows i j Left Velocity F (x, y = M (x, yi + N(x, y j (x, y Top Velocity F (x, y( j = N (x, y (x, y+ Right Velocity F (x, y(+i F (x, y( i = + M ( x, y F (x, y = M (x, y {N (x+, y N (x, y} = ( N x {M (x, y+ M (x, y} = ( M y (x+, y Bottom Velocity (x+, y+ F (x, y(+ j = + N(x, y Flow rate of counter clock wise circulating fluid Circulation around rectangle boundary ( N x ( M y = ( N x M y Circulation density = ( N x M y k-component Curl of F Circulation Density Divergence & Curl 22
23 2-D Curl (d Circulation around rectangle boundary ( N x ( M y = ( N x M y F (x, y Circulation density = ( N x M y k-component Curl of F Circulation Density Divergence & Curl 23
24 Clock-Wise & Counter-Clock-Wise Divergence & Curl 24
25 Clock-Wise & Counter-Clock-Wise X (i F (x, y = M (x, yi + N (x, y j N (x, y {N (x+, y N (x, y} N (x +, y N (x, y N (x +, y Divergence & Curl 25
26 Clock-Wise & Counter-Clock-Wise X (ii F (x, y = M (x, yi + N (x, y j N (x, y N (x +, y {N (x+, y N (x, y} > 0 > 0 {N (x+, y N (x, y} < 0 > 0 CCW bound net flow CW bound net flow N (x, y N (x+, y + + {N (x+, y N(x, y} N x > 0 CCW {N (x+, y N(x, y} N x < 0 CW Divergence & Curl 26
27 Clock-Wise & Counter-Clock-Wise X (iii F (x, y = M (x, yi + N (x, y j N (x, y N (x +, y {N (x+, y N (x, y} N x > 0 Positive Slope of a tangent line parallel to the x axis CCW bound net flow along the z axis {N (x+, y N (x, y} N x < 0 Negative Slope of a tangent line parallel to the x axis CW bound net flow along the z axis Divergence & Curl 27
28 Clock-Wise & Counter-Clock-Wise Y (i F (x, y = M (x, yi + N (x, y j M (x, y M (x, y + {M (x, y + M (x, y} M (x, y M (x, y Divergence & Curl 28
29 Clock-Wise & Counter-Clock-Wise Y (ii F (x, y = M (x, yi + N (x, y j M (x, y M (x, y + {M (x, y + M (x, y} > 0 > 0 {M (x, y + M (x, y} < 0 > 0 CCW bound net flow CW bound net flow M (x, y M (x, y + {M (x, y+ M (x, y} M y > 0 CCW {M (x, y+ M (x, y} M y < 0 CW Divergence & Curl 29
30 Clock-Wise & Counter-Clock-Wise Y (iii F (x, y = M (x, yi + N (x, y j M (x, y M (x, y + {M (x, y+ M (x, y} M y > 0 Positive Slope of a tangent line parallel to the y axis CCW bound net flow along the z axis {M (x, y+ M (x, y} M y < 0 Negative Slope of a tangent line parallel to the y axis CW bound net flow along the z axis Divergence & Curl 30
31 Clock-Wise & Counter-Clock-Wise Divergence & Curl 31
32 Clock-Wise & Counter-Clock-Wise N (x, y M (x, y M (x, y + N (x +, y {N (x+, y N(x, y} N x > 0 CCW {N (x+, y N(x, y} N x < 0 CW ( x i + y j + 0k (N j {M (x, y+ M (x, y} M y > 0 {M (x, y+ M (x, y} M y < 0 ( x i + y j + 0k (M i = ( N j = ( M i Divergence & Curl 32
33 2-D Curl and Del Operator 2-D Vector Field F (x, y = M (x, yi + N (x, y j Circulation around rectangle boundary ( N x ( M y = ( N x M y Circulation density = ( N x M y F (x, y = ( x i + y j + 0k (M i + N j + 0k Curl of F = F i j k 0 x y 0 M N Divergence & Curl 33
34 3-D Curl and Del Operator 2-D Vector Field Curl of F = ( N x M y k F (x, y = M (x, yi + N (x, y j i j k 0 x y 0 M N = ( x i + y j + 0k (M i + N j + 0k = F 3-D Vector Field F (x, y, z = M(x, y, zi + N (x, y, z j + P (x, y, zk Curl of F = ( P y N z i ( P x M z j + ( N x M y k i j k x y z M N P = ( x i + y j + z k (M i + N j + P k = F Divergence & Curl 34
35 2 i Divergence & Curl 35
36 2-D Divergence z F (x, y F (x, y+ F (x+, y F (x+, y+ y (x, y (x, y+ x (x+, y (x+, y+ Divergence & Curl 36
37 Chain Rule Function of two variable y = f (u, v u = g(x, y v = h(x, y Divergence & Curl 37
38 References [1] [2] [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] D.G. Zill, Advanced Engineering Mathematics
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