Surface Integrals (6A) Surface Integral Stokes' Theorem
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Arc Length In the Plane = f () b s = a 1 + ( d ) 2 d d d s d d Surface Integral 3
Surface Area In the Space z = f (, ) v Area of the surface over u Δ T A(S) = 1 + [ f (, ) ] 2 + [ f (, )] 2 d A = 1 + ( ) 2 + ( ) 2 d A Δ Δ Δ Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Δ T Surface Integral 4
Differential of Surface Area (1) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Slope along direction = 0.4 Slope along direction = 0.2 Δ u Δ T v Δ Δ u v Δ u = Δ i + Δ k Δ Δ Δ i k j v = Δ j + Δ k Surface Integral 5
Differential of Surface Area (2) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A Slope along direction = 0.4 Slope along direction = 0.2 Δ Δ Δ u v Δ Δ u Δ u = Δ i + Δ k i k j Δ Δ v = Δ j + Δ k Surface Integral 6
Differential of Surface Area (3) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A u = Δ i + Δ k v = Δ j + Δ k Δ u v Δ u Δ v Δ u v Δ T i j k Δ 0 u v = Δ 0 Δ Δ = [ i j + k ] Δ Δ i k j u v Δ T u v = ( ) 2 + ( ) 2 + 1 Δ Δ Surface Integral 7
Differential of Surface Area (4) Differential of surface area ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A u = Δ i + Δ k v = Δ j + Δ k Δ u v Δ u Δ v Δ Δ T u v = [ i j + k ] Δ Δ u v = ( ) 2 + ( ) 2 + 1 Δ Δ S i k j Δ Δ Δ Δ T = [ f (, )] 2 + [ f (, )] 2 + 1 Δ Δ Δ S = [ f (, )] 2 + [ f (, )] 2 + 1 Δ A Surface Integral 8
Line Integral with an Eplicit Curve Function = f () a b Curve C d d = f '() d = f '() d d s = [d ] 2 + [d ] 2 d s = 1 + [ f '()] 2 d d s d d C G(, ) d = a b G(, f ()) d C G(, ) d = a b G(, f ()) f '() d C G(, ) d s = a b G(, f ()) 1 + [ f '()] 2 d Surface Integral 9
Surface Integral with an Eplicit Surface Function z = f (, ) a b c d egion d f d = f (, ) d f d = f (, ) ds = 1 + [ f (, )] 2 + [ f (, ) ] 2 d A S G(,, z) d S = G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A Surface Integral 10
Surface Integral Over XY (1) domain z ds 3 range 4 G(,, f (, )) z z = f (, ) G(,, z) G(,, f (, )) a b c d Surface Integral 11
Surface Integral Over XY (2) domain z ds 3 range 4 G(,, f (, )) z z = f (, ) S G(,, z) d S = G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A Surface Integral 12
Surface Integral Over XZ (1) domain z 3 range 4 G(, g(, z), z) z ds = g(, z) G(,, z) G(, g(, z), z) a b c d Surface Integral 13
Surface Integral Over XZ (2) domain z 3 range 4 G(, g(, z), z) z ds = g(, z) S G(,, z) d S = G(, g(, z), z) 1 + [g (, z)] 2 + [g z (, z) ] 2 d A Surface Integral 14
Surface Integral with an Eplicit Surface Function z = f (, ) egion S G(,, z) d S = g(, z) egion S G(,, z) d S = h(, z) egion S G(,, z) d S d f d = f (, ) d f d = f (, ) = = ds = 1 + [ f (, )] 2 + [ f (, )] 2 d A G(,, f (, )) 1 + [ f (, )] 2 + [ f (, )] 2 d A d g d = g (, z) d g d z = g (, z) = ds = 1 + [g (, z )] 2 + [g z (, z )] 2 d A G(, g(, z), z) 1 + [g (, z)] 2 + [g z (, z) ] 2 d A dh d = f (, z) d h d z = f z (, z) ds = 1 + [h (, z) ] 2 + [h z (, z)] 2 d A G(h(, z),, z) 1 + [h (, z)] 2 + [h z (, z)] 2 d A Surface Integral 15
Mass of a Surface densit ρ(,, z) mass m = S ρ(,, z) d S Surface Integral 16
Orientation of a Surface Oriented Surface there eists a continuous unit normal vector function n defined at each point (,,z) on the surface Orientation the vector field n(,, z) or n(,, z) Upward orientation Downward orientation z = f(,) : positive k component z = f(,) : negative k component Surface g(,, z) = 0 Unit Normal Vector n = 1 g g g = g i + g j + g z k Surface z = f (, ) g(,, z) = z f (, ) = 0 g(,, z) = f (, ) z = 0 Surface Integral 17
Level Curve and Surface Functions of two variables z = F (, ) 3 G(,, z) = 0 Level Curve c i = F (, ) 2 F (, ) = 0 when z = 0 0 = F (, ) r '(t) Functions of three variables w = G(,, z) 4 H (,, z,w) = 0 Level Surface c i = G(,, z) 3 G(,, z) = 0 when w = 0 0 = G(,, z) r '(t) Surface Integral 18
Gradient & Normal Vector Functions of three variables w = F (,, z) 4 Level Surface c 0 = F (,, z) dc 0 d t = d F dt 0 = F (,, z) d dt + F d dt + F z 3 d z dt = f (t) = g(t) z = h(t) 0 = ( F i + F j + F z k ) ( d dt i + d d t j + d z d t k ) 0 = F (,, z) r '(t) tangent vector F normal to the level surface at P ( 0, 0, z 0 ) Surface Integral 19
Surface Integral over a 3-D Vector Field (1) A given point in a 3-d space ( 0, 0, z 0 ) A vector P ( 0, 0, z 0 ), Q( 0, 0, z 0 ), ( 0, 0, z 0 ) domain range 3 functions z ( 0, 0, z 0 ) z n F ( 0, 0, z 0 ) ( 0, 0, z 0 ) P ( 0, 0, Z 0 ) ( 0, 0, z 0 ) Q( 0, 0, z 0 ) ( 0, 0, z 0 ) ( 0, 0, z 0 ) ( 0, 0, z 0 ) onl points that are on the surface F ( 0, 0, z 0 ) = P ( 0, 0, z 0 )i + Q( 0, 0, z 0 ) j + ( 0, 0, z 0 )k consider onl the component of F along n F n Surface Integral 20
Surface Integral over a 3-D Vector Field (2) F (,, z) = P(,, z) i + Q(,, z) j + (,, z) k Line Integral over a 3-d Vector Field r (t) = f (t) i + g(t) j + h(t) k c F dr = C P(,, z) d + Q(,, z) d + (,, z) d Surface Integral over a 3-d Vector Field n = g g (F n) ds S = flu total volume of a fluid passing through S per unit time F velocit field of a fluid Surface Integral 21
Vector Form of Green's Theorem A force field A smooth curve Work done b F along C C F (, ) = P (, )i + Q(, ) j C : = f (t), = g(t), a t b W = c F dr = c F T d s = C P(, ) d + Q(, ) d C P d + Q d = ( Q P ) d A curl F = F = i j k = z P Q 0 ( Q P ) k ( F ) k = ( Q P ) C P d + Q d = ( F ) k d A Surface Integral 22
Stokes' Theorem (1) A force field F (, ) = P (, )i + Q(, ) j Work done b F along C W = c F dr = c F T d s n C T v 2-space = ( F ) k d A 3-space = ( F ) n d A Surface Integral 23
Stokes' Theorem (2) A force field F (, ) = P (, )i + Q(, ) j Work done b F along C W = c F dr = c F T d s n T 3-space = ( F ) n d A i j k curl F = F = z P Q v = ( Q ) z i + ( P z ) j + ( Q P ) k n = Surface Integral 24 i j + k 1 + ( ) 2 + ( ) 2 surface g(,, z) = z f (, )
Gradient of a 2 Variable Function Function of two variables f (, ) f (, ) = i + j vector ate of change of f in the direction ate of change of f in the direction Slope in the direction = 2 Slope in the direction = 1 z k j = 1 j i = 2i i θ j u Surface Integral 25
2-D Divergence z F (, ) F (, +Δ ) F (+Δ, ) F (+Δ, +Δ ) (, ) (, +Δ ) (+Δ, ) (+Δ, +Δ ) Surface Integral 26
Chain ule Function of two variable = f (u, v) u = g(, ) v = h(, ) Surface Integral 27
eferences [1] http://en.wikipedia.org/ [2] http://planetmath.org/ [3] M.L. Boas, Mathematical Methods in the Phsical Sciences [4] D.G. Zill, Advanced Engineering Mathematics