Divergence and Curl (3B Divergence Curl Green's Theorem
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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
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
Inward & Outward Bound Divergence & Curl 5
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
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
Inward & Outward Bound Divergence & Curl 8
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 3 +3 +3 3 M x + + + + Divergence & Curl 9
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 3 +3 +3 3 M x Divergence & Curl 10
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 3 +3 +3 3 {M (x+, y M (x, y} M x < 0 Inward Divergence & Curl 11
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
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 3 +3 + + +3 3 + Divergence & Curl 13
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 3 +3 +3 3 Divergence & Curl 14
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
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
Inward & Outward Bound Divergence & Curl 17
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
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
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
Divergence & Curl 21
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
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
Clock-Wise & Counter-Clock-Wise Divergence & Curl 24
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 + + 3 + 3 + +3 +3 Divergence & Curl 25
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 + 3 +3 + 3 +3 {N (x+, y N(x, y} N x < 0 CW Divergence & Curl 26
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
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 + + + + + +3 3 3 +3 Divergence & Curl 28
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 + + + + +3 3 3 +3 {M (x, y+ M (x, y} M y < 0 CW Divergence & Curl 29
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
Clock-Wise & Counter-Clock-Wise Divergence & Curl 31
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
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
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
2 i Divergence & Curl 35
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
Chain Rule Function of two variable y = f (u, v u = g(x, y v = h(x, y Divergence & Curl 37
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