IV.3 VECTOR ANALYSIS Revisited and Enhanced

Transcription

1 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 69 I. ETOR NLYI Revisited and Enhanced

2 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 7. ETOR FUNTION Let be a vecto in the Euclidian vecto space. Then the vecto function of a scala vaiable is a vecto-valued function defined as a map fom the set of eal numbes to the space of vectos : ( t) t t z t t :I (4) The scala vaiable t I can epesent time, path, etc. The vecto function can povide a convenient method fo the definition of cuves in space b tacing t, t a,b. The change of paamete the points b the position vecto [ ] t [ a,b] also povides infomation about the position of the point on the cuve fo diffeent moments of time. This definition of cuves b vecto functions is equivalent to the paametic definition of cuves: t ( t) t o z t whee ( t ),( t ),z( t ), t [ a,b] t t z z t ae eal valued functions. [ a,b] t (4) LINE IN PE Let the point P (,,z) some vecto u ( u,u,u ) be defined b the position vecto and let u be. onside a line L which goes though the point P in the diection of vecto u. n vecto in the line L is collinea to vecto u and theefoe is a multiple of the vecto u with some coefficient tu. Then the position vecto defining all points on the line L can be epesented as a vecto function of the vaiable t : L ( t) + tu t (44) In the component fom this function is given b the equation P u ( t) + tu t u t t + t u t (45) z t z u which coesponds to a paametic definition of a line: t + tu t + tu z t z + tu t (46) The othe equation which detemines the coodinates (,,z ) of the points on the line L can be obtained b solving each of the paametic equations fo t : z z t u u u povided that all u. Then thee equations specif the points of the line L : z z (47) u u u These ae called the smmetic equations of a line.

3 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 7 P u P ( t) L Let us deive now the equation of the line which goes though two given points P (,,z) and P (,,z) detemined b the position vectos and. The diection vecto fo this line can be found as u OP OP,,z z ccoding to Equation (44), the line is defined b the following vecto function: u ( t) + tu t u t ( ) t t (48) z u z z z P P The segment connecting the points P (,,z ) and P (,,z ) defined b the equation: can be ( t) ( t) + t( ) t + ( t) t (49) PLNE IN PE n Π Geometicall a plane can be defined b seveal diffeent was the ield diffeent vecto equations defining a plane: ) The plane Π can be uniquel defined b the fied point (,,z) and a vecto n ( n,n,n) othogonal to the plane Π : Let vecto (,,z) define an abita point on the plane Π, then the vecto belongs in the plane Π. Hence all vectos on the plane Π ae othogonal to the vecto n, the vectos ae othogonal to n. Theefoe, n (5) Using the distibutive law of dot poducts, one obtains the vecto equation of a plane n c (5) whee c n n + n + nz. In the coodinate fom, this equation can be witten as n + n + nz n + n + nz (5) We can notice, that if the plane is defined b the equation a + a + az c c (5) then the vecto a ( a,a,a) is othogonal to this plane. The planes defined b Equation (5) fo diffeent values of the coefficient c ae paallel to each othe. Equation (5) defines coodinates (,,z ) which belong to the plane.

4 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 7 a b b a Π ) The plane Π can be uniquel defined b the fied point (,,z) and two vectos a ( a,a,a ) and b ( b,b,b ) ling on the plane Π. The vecto obtained b coss poduct a bis othogonal to the plane Π. Then the vecto equation defining the plane Π accoding to Equation (5) is which ields ( ) ( a b ) a b a b (5) In the coodinate fom this equation can be witten as ( ) ( a b ) i j a a a z z b b b ( - )( ab ab ) ( - )( ab ab ) ( z-z )( ab ab ) i j+ a b b a Π ) The plane Π can be uniquel defined b thee fied points (,,z), (,,z), and (,,z) : ectos a (,,z z) and b (,,z z ) ae ling in the plane Π. Then fom Equation (5), it follows that (54) defines the plane Π. The left hand-side of the equation (54) is a tiple scala poduct, which accoding to () can be witten as z z z z z z (55) PHERE IN PE a spheical suface of adius a with the cente at the oigin is a set of all points witten in the vecto fom as the distance fom which to the oigin is equal to a. This condition can be a a Then the vecto equation of the sphee of adius a with the cente at the oigin is a (56) The equation of a sphee of adius a with the cente at the point (,,z ) can be deived in the following fom ( ) ( ) a EXMPLE:. Find the distance fom the point (,,z ) (57) to the plane n c.

5 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 7 d n n c Because the plane is defined b the vecto equation n c, vecto n is,,z othogonal to the plane. Then a line going though the point in a diection of the vecto n is also othogonal to the plane: + n t This line intesects the plane at the moment of time t fo which the equation of the line defines the point belonging to the plane n c : + t n n c n+ t nn c n + t c t c Then the point of intesection is defined b + t n + c nn The vecto d n n c n is a fee vecto connecting the points and and it is othogonal to the plane. Theefoe, the nom of d defines the distance fom to the plane: d n c n n c n n c (58) v + tv t t t n c. Let the paticle be taveling along the line with the constant velocit v. If at t,,z, detemine at which the paticle was at the point moment of time it cosses the plane Π : n c. The equation govening the motion of the point in the diection of the velocit vecto v along the line though the point is + tv Intesection of the plane n c ields ( + tv) n c Then, using the distibutive popet of dot poduct n+ tvn c tvn c n c n t vn (59) a. Deive the equation of the plane tangent to the sphee a at. Because the point belongs to the sphee, its nom is equal to a and it satisfies the equation a The plane contains the point and it is othogonal to the vecto. Then the vecto equation of the plane accoding to (5) is a Using the distibutive popet of dot poducts, we can obtain a Then the vecto equation of the tangent plane becomes: a (6) It can be ewitten in the following coodinate fom + + zz a

6 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, DIFFERENTITION ND INTEGRTION OF ETOR FUNTION pace uve onside a bounded space cuve in with the end points and B. The gaph of the space cuve is taced b a vecto function : ( t) t [ a,b] with ( a ), b B (6) Paametic definition of the cuve ( t) ( ( t ),( t ),z( t) ) : t ( t) ( t) ( t) t [ a,b] z z( t) ( t) : (6) ontinuit ontinuit of the vecto function at the moment of time t : ( t) ( t+ t) ( t) ( t+ t) t ( + ) lim t t t (6) This condition is equivalent to the following condition: t ( ) lim t + t t (64) Deivative d dt ( t) The deivative of a vecto function with espect to a scala vaiable is defined as ( t+ t) ( t) d ( t) ( t) lim (65) dt t t If the limit eists, then the deivative also is a vecto function d dt ( t ) : ( t) ( t+ t) The vecto d dt is on the line tangent to the cuve at the point ( t). In paametic fom, the diffeentiation of a vecto function educes to diffeentiation of its components: t ( t) ( t) ( t) ( t) ( t) + ( t) + z ( t) z( t) z ( t) Repeatedl, the highe ode deivatives can be defined: d ( t) ( t) dt d d d ( t) ( t) ( t) dt dt dt d d d ( t) ( t) ( t ) dt dt dt i j (66)

7 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 75 Mechanical sense of the vecto deivatives: ( t) a( t) v ( t) v ( t+ t) v ( t) ( t t) ( t) + is displacement of the point in time t t t is the aveage velocit ove time lim ( t) v ( t) is the instantaneous velocit at time t t ( t t) ( t) v v + v is the change of velocit in time t v lim t t t t v a is the acceleation at the time t, a( t) v ( t) ( t) If the motion of the point is defined b its components ( t) ( ( t ),( t ),z( t) ) then ( t) ( t) + ( t) + z( t) i j ( t) ( t) + ( t) + z ( t) v i j ( t) ( t) + ( t) + z ( t) a i j t, diffeential element on the space cuve is tangent to the cuve: d ( d,d,d ) d d ds ds T d T ds T (67) dt ds dt dt PROPERTIE Let u ( t) and ( t). ( cu) v be diffeentiable vecto functions, then: cu u u u. f ( t) f ( t) + f ( t). ( u+ v) u + v 4. ( u v) u v+ u v 5. ( u v) u v+ u v d da du dv + + dt a u v u v a dt v dt a u dt 6. d da du dv + + dt a u v u v a dt v dt a u dt 7. d d ds dt (chain ule) ds dt 8. s( t)

8 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 76 how how the popet () can be deived: f t u d f ( t) lim dt u t ( + ) u( + ) u f t t t t f t t t + u + u u f t f t f t t lim t t fu+ f u+ f u df du lim u + f t t dt dt In a simila manne the othe popeties ae deived. TYLOR ERIE If the components of vecto function ( t) ( ( t ),( t ),z( t) ) ae epandable into a Talo seies ove the point t t then the vecto function ( t) also can be epanded into a Talo seies: Talo eies ( t t ) t t t t t t t t t... (68)!!! INDEFINITE INTEGRL The vecto function a ( t) is the antideivative of the vecto function ( t) d a dt ( t) ( t) if Then the indefinite integal of a vecto function is the opeation of finding all antideivatives, which is denoted in the taditional wa: ( t) dt t dt t dt i+ t dt j+ z t dt ( t) dt a( t) + c z ( t) dt Indefinite Integal whee the constant vecto c is a constant of integation. DEFINITE INTEGRL Definite integal of the vecto function ( t) can be defined as a limit of Riemann s sum v u n ( t) dt lim ( ti)( ti ti ) b n i whee { u t,t,...,t v} is a patition of the inteval [ ] n u,v. The Fundamental Theoem of alculus fo vecto functions is fomulated in t t then a simila wa: if a is the antideivative of a vecto function

9 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 77 Definite Integal v ( t) dt u v v b t dt ( t) dt a( v) a ( u) (69) u u v z ( t) dt u LENGTH OF PE URE Deivation of these esults can be pefomed fo the component fom of the t t, t,z t based on the integation of the eal- vecto function ( ) valued functions of a single vaiable ( t ),( t ),z( t ). s ( b) If a space cuve is defined paameticall b a vecto function t t, t,z t, then the length of cuve (ac length) is detemined ( ) b the integal b a b a s t dt t + t + z t dt (7) ( a) PRMETRIZTION OF PE URE IN TERM OF R LENGTH Natual paametization: the pathlength s is used as a paamete fo the vecto function: ( s) s [ a,b] s s ( s) T ( s) t t + + a a s t t dt t t z t dt olve fo t in tems of s, substitute into ( t) ( ( t ),( t ),z( t) ) d s ( t ) ( t ) ( t ) + ( t ) + z ( t ) dt. The scala vaiable in the vecto function in tems of ac length: s( t), ( s ) s s,s d d ds d s( t) ( t) dt ds dt ds s a s t dt d s ds d ds s, [ ], s s( t ) ( t) dt ( s) Theefoe, ( s) a is alwas a unit tangent vecto. It is denoted b d v T ( s) ( s) ds v

10 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 78 NORML PLNE TO THE URE Nomal plane to the cuve T ( t) ( ( t ),( t ),z( t) ) at the point ( t ). ( t) The nomal unit vecto T ( t ) ( t ) is othogonal to the plane. The equation of the plane though the point ( t ) nomal to T (5): T (7) Definitions of some paticula space cuves: is a closed cuve if ( a) ( b) o ( a) ( b) is a smooth cuve if ( t) [ a,b] i i (deivatives ae continuous) i is piece-wise smooth if whee ae smooth cuves is a simple cuve if ( t) ( t ) if t t (w/o selfintesection) i i EXMPLE: The gaph of the space cuve taced b the vecto function t cos t,sin t,t sin t i+ cos t j+ t t z t π Fo discete values of the paamete t, the coodinates of points taced b a vecto function can be calculated fom the given equation: t z ( t) t π π 4 π 4 π π π π t t π 4 and the cuve can be dawn though these points. The peiodic functions defining the and the coodinates ield the cicula pojection of the space cuve on the -plane: + and the linea incease of the z-coodinate ields the spial shape of the space cuve.

11 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, LR ND ETOR FIELD We consideed vecto functions of the scala vaiable (time). Now, we intoduce functions of the vecto vaiable (point in space). ϕ ( ) Let be a position vecto which specifies a location in Euclidian space. scala field is defined as a eal valued function of a vecto vaiable: : ϕ ϕ (,,z ) : B this function, a scala value is specified fo an point of space. scala field can descibe distibution in space of tempeatue, densit, concentation, etc. ϕ ϕ, : defines a scala field on a plane. Function v( ) vecto field is defined as a vecto valued function of a vecto vaiable: v( ) : P(,,z) + Q(,,z) + R(,,z) v i j B this function, a vecto value is specified fo an point of space. vecto field can be descibed b a distibution in space of velocit, acceleation, foce, etc. v(,t) non-stationa scala o vecto field ae defined as time-dependent maps (,t ) : ϕ (,t) : v v( ) v( ) v v( ) ll consideed vecto functions ae assumed to be continuous: lim ϕ lim ( ) ϕ( ) v( ) v( ) lim ( ) v v ll opeations defined fo vectos can be applied fo vecto fields point-wise.

12 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 8 LEEL URE ND URFE f : be a scala field, then equation Let f c o f (,) c descibes the cuves in the plane called the level cuves: {(, ) f (, ) c, c } Isothems: {(,) T (,) c} Level cuves ae descibed b the functions of two vaiables f (, ). ϕ : be a scala field. Then equation Let descibes the suface in ϕ ( ) c o (,, ) ϕ c called the level suface: {(,, ) ϕ (,, ) c, c } Isothemal uface: {(,,z) T (,,z) T } Fo diffeent values of the constant c, we obtain the families of un-intesected level cuves and sufaces (wh do level cuves not intesect?). Level sufaces ae descibed b the functions of thee vaiables (,, ) ϕ. OPERTOR NBL, GRDIENT, DIRETIONL DERITIE nabla : is a diffeential vecto opeato called nabla and defined as,, z (7) ometimes it is also called the Hamilton opeato. gadient Opeato nabla applied to a scala-valued vecto function ϕ ields a vecto called the gadient of the scala field gadϕ ϕ ϕ, ϕ, ϕ (7) ϕ + ϕ i i + ϕ i (7) which is a vecto othogonal to the level suface of the scala field o to the level cuve in the -D case.

13 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 8 Eample: Let the position vecto (,) ϕ define points on the plane. ϕ ϕ, +. nd let a scala field be defined b This scala field can be visualized b the level cuves + c which ae concentic cicles aound the oigin. The gadient of this scala field is + c ϕ (,) ( + ) i+ ( + ) j i+ j (,) o in vecto fom it can be witten as ϕ ( ) Popeties of gadient:. ( ϕ+ ψ ) ϕ+ ψ. ( c ϕ ) c ϕ, c. ( ϕψ ) ψ ϕ + ϕ ψ Diectional deivative of the scala field The gadient at the point (,) passing though this point. The point P (,) belongs to the level cuve The value of the gadient at this point is ϕ, i+ j i+ 4j,4 is nomal to the level cuve + 5. Let the function ϕ : define a scala field, and let s be a unit vecto, s. ecto hs with a small h > is an incement in the diection s. Then a deivative of the function is defined as D ϕ s ϕ lim h in the diection s at the point in space ( + hs) ϕ( ) It detemines the ate of change of the scala field diection s. ϕ s h at the point in the Using the diffeentiation ules fo the multivaiable functions and the definition of the opeato nabla, let us wite the othe epesentations of the diectional deivative: ϕ ϕ + ϕ + ϕ s s s s ϕ ϕ ϕ cos, + cos, + cos, ( si ) ( si ) ( si ) ϕ ϕ ϕ s+ s+ s φ si i ϕ ϕ ( ) ϕ ( ) (74) s (75)

14 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 8 Theefoe, the deivative of ϕ in an diection is equal to the pojection of the gadient onto this diection: ϕ ϕ s ϕ s ϕ cos ( ϕ, s ) (76) It follows fom this equation that the maimum value of the diectional deivative is achieved in the diection of the gadient of the scala field at this point. o we can conclude that the gadient of the scala field is a vecto which has a diection of the geatest incease and its magnitude is equal to the diectional deivative in this diection. The opposite diection ϕ coesponds to the diection of the geatest decease. Fom equation (76) also ields that dϕ ϕ d If n is the unit nomal vecto to the level suface of ϕ, then ϕ ϕ n (77) n Diectional deivative of the vecto field Let the function a : define a vecto field, and let s be a unit vecto, s. ecto hs with a small h > is an incement in the diection s. Then the deivative of the function a in a diection s at the point of space is defined as D s a a a ( + hs) a s lim h h (78) povided that the limit eists. It detemines the ate of change of the vecto field a at the point in the diection s : s a a a a + + s s s a a a cos, + cos, + cos, ( si ) ( si ) ( si ) a a a s+ s+ s ( sa ) (79)

15 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, DOUBLE INTEGRL onside the function of two vaiables f f ( ) f ( ) Definition of the double integal: f ( ) d ( ) σ defined in. lim f (8) alues of scala function f {(,,z) (,),z f (,) }, then is a pojection of on -plane. define the suface: Geometical sense: if f (,) ea of d (8) atesian coodinates: (,) : f ( ) d f if f (,) olume between and f, d lim f (, ) σ f d (8) f, dd f f, { } I (, ) a b, g ( ) g ( ) Iteated integal: f (, ) d f (, ) d d (8) b g ( ) a g ( ) { } II, h h, c d Iteated integal: f (, ) d f (, ) d d (84) d h ( ) c h ( ) Pola coodinates: (, θ ) : f ( ) d f (, θ ) ( θ) f, ddθ f f,θ f + + d lim f (, θ ) θ σ { } I (, θ) a b, g ( ) θ g ( ) Iteated integal: f (, θ ) d f (, ) dθ d (85) b g ( ) a g ( ) θ { θ θ θ α θ β } II, h h, Iteated integal: + θ + + θ θ + f (, ) d f (, θ) d dθ (86) β h ( θ ) α h ( θ)

16 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 84 RTE OF HET TRNFER Rate of Heat Tansfe though the uface : (87) q q (, ) d whee q (,) is the local heat flu on the suface in a diection of ais z (nomal heat flu) ERGING veaging of cala Field z f (,) on the uface : (88) T av T, d aveaged tempeatue on (89) h h, d h(, ) is the local convective coefficient on Eample: Given the heat flu in z diection though the aea { },, 5 + (nomal distibution of heat flu ove aea ): q (,) W 6 m What is the ate of heat tansfe q though the aea? What is the aveaged value of the heat flu on? W q (, ), m q The ate of heat tansfe: q q +, d + q (, ) d d d d + 6 d d d 5d W + 5 d ea: veaged value: q av + d d 5 [ ] d m q, d W 99.8 m

17 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 85 Eample: onside a coss-section of a cicula pipe of adius R. Let the tempeatue in the coss-section be defined b T (,) Find the mean tempeatue T m of the coss-section. T (,) + 4 T, ( θ ) T, + 4 T m T ( ) d πr 4 π onvesion to pola coodinates: T (, θ ) cosθ + 4 { } sinθ (, θ), θ π + T m π ( θ) T, ddθ π dd π d π 4π d ( + ) d ln 4 + θ 5 ln.7

18 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 86 URFE RE onside a suface defined b the equation, z f (,), onside a patition of aea onto small aea incements ea incements in the -plane define the aea incements of the suface f: onside a point (,,z f (,) ) on the suface. onstuct a tangent plane to the suface at this point, and then the pojection T of the aea and coespondingl of onto the tangent plane. Fist find the vectos u and v z z T f u u { i+ z i+ f z f, z f (,) i f u i+ z i+ i+ f, f v j+ z j+ j+ f, ea u v T i j det f f,, f i f j+,, f i f j+,, f + f +,, Then the aea of the suface can be appoimated as: T f + f +,, In a limit, with the nom of patition appoaching zeo, the aea is defined b the double integal: lim f + f + σ,, f + f + dd f f d (9) + +

19 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 87 : Repesentation of suface integal of scala function with diffeent choice of pojection of the suface z z f (,) lim f + f + σ,, f f dd (9) + + z f (,), (,) z g(,z) z lim g + g + z σ, z, g gz ddz (9) z + + g(,z), (,z) z z z h (,z) lim h + h + z σ, z, h hz ddz (9) z + + h(,z), (,z) z

20 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, URFE INTEGRL OF THE LR FUNTION onside the scala function defined on the suface : G G(,,z ) : Let the suface be defined b the equation: g,,z Then the nomal vecto to an point on the suface is defined b g n g ssume that the suface allows also epesentations: z f,,, g(,z), (,z) z h(,z), (,z) z onstuct a suface G. What we ae going to obtain with the suface integation of scala function G is the aea between sufaces and G (with positive sign if G is to diection of n, and minus sign if it is in the opposite diection): G,,z d lim G σ lim G f + f + σ,, G (,,z) d G,, f, f f dd (94) + +, G (,,z) d G (,,z) d G,g,z,z g gz ddz (95) z + + G h,z,,z h hz ddz (96) z + +

21 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 89 Eample: Find lateal suface aea of a cone with h and R : h z f (,) + + h equation of the suface R h s f f (,) (,) h R + h R + f f h R + (,) h R + (,) R f + f + dd R R conside one quate of a cone 4 f + f + dd h 4 + dd R R integate with Maple R T.66 π R R + h π Rs T.85 Let h, R Tempeatue distibution: T (,,z) + + z + T.6 uface aveage tempeatue: T ave? T ( R,R,) π R R + h.57 suface aea z T ave T,,z d T,, f (, ) f + f + dd T (,,) R R 4 T,, f (, ) f + f + dd z.78 integated with Maple Rema: integation can be simple in clindical coodinates

22 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, URFE INTEGRL OF THE ETOR FUNTION FLUX OF ETOR FIELD g(,,z) Let the suface be defined b the equation: g,,z The unit nomal vecto to suface is defined b g n g (fo the diection of the nomal vecto, we will agee to tae the eteio diection fo closed sufaces, and one of two diections fo non-closed sufaces and stic to this diection when changing position on the suface). ssume that the suface allows epesentations: a, be defined b Let the vecto field The dot poduct a n z f (,), (,) g(,z), (,z) z h(,z), (,z) z a a + a + a i j a i+ a j+ a z a,,z i+ a,,z j+ a,,z z an a cos ( ni, ) + a cos ( ni, ) + a cos ( ni, ) a n + a n + az nz is the magnitude of the pojection of vecto a in the nomal diection n. ubdivide suface onto subsufaces with aeas which can be assumed to be flat and be chaacteized b the nomal vecto with the magnitude equal to the aea : n. Then the suface integal can be defined as a limit of the sums Using the diffeential elation d n d, we can epess a d ( an )d and a d lim a ( ni ) ( ni ) ( ni ) a cos, a cos, a cos, + + d ( ni ) ( ni ) ( ni ) a cos, d + a cos, d + a cos, d a d d + a d d + a d d (97)

23 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 9 FLUX Let us define a flu of a vecto field a though the suface as a suface integal of the vecto function a : { } flu of a though the suface a d (98) In the Heat Tansfe theo, W flu of the vecto field q m (which is defined as a heat flu itself) though the suface [ ] defines the ate of heat tansfe q W though the suface (if it is positive, then the heat tansfe occus fom the suface in the diection of the nomal vecto n; and if it is negative, then it occus in the opposite to nomal vecto diection). Theefoe, flu of q though the suface is the ate of heat tansfe though [ ] q q W Using Equation (8), the flu of the vecto field though the suface can be witten in the taditional fom of the suface integal: a d an d (99) Evaluation of the suface integal can be pefomed with the help of an fom of equation (97). If we denote scala poduct b the scala function G ( ) G(,,z) an then the flu of the vecto field can be calculated using one of the equations (94-96), if the coesponding epesentation of the suface (p.4) is allowed: a d G (,,z) d a d G (,,z) d G,, f, f f dd () + + a d G (,,z) d G,g,z,z g gz ddz () z + + G h,z,,z h hz ddz () z + + ll thee foms of the suface integal with the help of components of scala poduct of vecto field with the suface nomal vecto an a n + a n + az nz G,,z + G,,z + G,,z z ae taditionall combined in a single equation [Kon&Kon, p.56]: a d G,, f (, ) f + f + dd G,g,z,z g gz ddz () z z + Gz h,z,,z h + hz + ddz The flu though the closed suface is denoted b a d an d (4) Useful fact: the flu though an closed suface in the constant vecto field a ais zeo (consevation law): a d

24 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 9 Eample: (Zill, p.59, m solution) Find the flu of the vecto field a,,z zj+ z (vecto field) z+ z 6 z z F n + z+ though the suface defined b the plane + + z 6 in the fist octant oiented upwad. uface is defined altenativel b the following equations: g(,,z) + + z 6 z f (,) + 6 f f g(,z) z + g gz h(,z) z + h hz Unit nomal vecto to suface : g n (,,) i+ j+ g cala field G(,,z) is defined b a scala poduct of the vecto field with the nomal vecto: G,,z an alculate the flu using equation (): G G Gz + z+ z G z 4 a d d an G (,,z) d G,, f, f + f + dd - + ( + 6 ) ( ) + ( ) + dd ( + 6 ) 4dd [ ] + 6 dd d d d d

25 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 9 Othe solution: alculate the flu using equation (): a d G,, f, f + f + dd z + G,g,z,z g + gz + ddz z + Gz h,z,,z h + hz + ddz z + g + gz + ddz 4 z z + h + hz + ddz 4 z z 9 ddz z + z 4 ddz z z + 6 zddz z zddz 6 z z + dz 6 z + z dz + 6 z + z dz 6 z + z dz + z + z z z nswe: Rate of heat tansfe though the suface : q n a d 8

26 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 94 Eample: z onside the long ectangula column with the nown tempeatue distibution 5 T ( ) T (,,z ) : scala field T (,,z) T,,z isothems K q? M n L T (,,z) L 4. M. K. themal conductivit Descibe and visualize the heat tansfe in the column, and detemine the ate of heat tansfe though the suface { } L,,z M, z K. Tempeatue scala field defines the gadient vecto field T (,,z) T T T i+ j+ 5 i Gadient vectos ae othogonal to isothems, the ae in the diection of geatest incease of tempeatue. Gadient vectos ae tangent to the lines of heat flow. Gadient vecto field defines the heat flu vecto field b Fouie s Law: j q (,,z) T (,,z) i+ ( + + 4) ( + + 4) j Heat flu vectos ae in a diection of geatest decease of tempeatue. gadient vecto field T (,,z) heat flu vecto field q (,,z) isothems M n q ate of heat tansfe though the suface in a diection of n L

27 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 95 uface is in the plane defined b equation g(,,z) L Equation fo defined b pojection on z plane (p.4): L h(,z), M, z K Unit nomal vecto to suface : n i+ j+ i h hz The scala field defined b the pojection of the heat flu on nomal diection: Gn (,,z ) q (,,z) n ( + + 4) Rate of heat tansfe though the suface : q d n q n G (,,z) d G h,z,,z g + gz + ddz z KM L ( L + + 4) ( + ) ddz 4 ddz 4 d.79 ( + ) nswe: Rate of heat tansfe though the suface : q n a d.79

28 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, DIERGENE Divegence of the vecto field a at the point of space is defined as a limit of the aveaged flu though the suface of the abita volume containing point : diva lim a d (5) Use a paallelepiped fo the abita volume with one cone located at the point (,,), sides i and faces pependicula to the coodinate aes, and with the sufaces, (see pictue). Then diva lim a d an d lim a lim a n lim ( ),,,,... lim a + a i + lim lim lim lim a +,, a,, i +... ( ) a +,, i a,, i +... ( ) a +,, a,, +... ( ) a +,, a,, +... a a a + + a (6) Fo incompessible fluid, the divegence of the velocit vecto field is zeo: div v

29 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, URL The cul of the vecto field a at the point of space is defined as a limit of the aveaged flu though the suface of the abita volume containing point : cula n a d lim (7) ompae with divegence (equation (6)): diva na d lim a (8) pplication of the opeato nabla ields a simila epesentation of the cul cula a (9) Useful fomulas: cula a i i i a a a () The components can be detemined b a a j ( cula ) i j whee indices i, j, ae a cclic pemutation of the numbes,,. Fo iotational fluid, the cul of the velocit vecto field is zeo, cul v ϕ ϕ ϕ + + z Laplacian Opeato ϕ ϕ ϕ div( gadϕ ) : ϕ IDENTITIE: Let a, b : be vecto fields, ϕ :. cul ( gadϕ ) be scala field, c ϕ. div( cul a) a. div( a+ b) diva+ divb 4. div( ϕa) ϕdiva+ a gadϕ a+ b a+ b ϕa ϕ a+ a ϕ 5. div( a b) b cula a culb ( a b) b ( a) a ( b ) 6. div( ca) cdiva ( ca) 7. cul ( a+ b) cula+ culb c a a+ b a+ b ϕa ϕ a + ϕ a 8. cul ( ϕa) ϕcula+ gadϕ a 9. cul ( cula+ gadϕ ) cul ( cula ) ( a+ ϕ ) ( a ). cul ( cula) gad ( diva) a ( ) ( ) a a a

30 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, LINE INTEGRL onside a vecto-valued function a, whee vectos belong to the space cuve : ( t) t [ a,b] with ( a ), b B () We will conside a line integal which smbolicall is witten as: a d () Let us see how this integal is defined in its phsical sense. et up a patition P n of the cuve into a discete set of n points: { } P a,,,..., b n n and define the incement Define the nom of patition as the biggest incement in the patition: P ma Denote the values of function Fom a dot poduct n a at the points of patition which has a phsical sense of the wo pefomed b the foce a ove path. Then the line integal is defined similal to the definition of the definite integal as a limit of the Riemann sum: which phsicall epesses the wo done b the foce a cuve. ( ) i ( ) j ( ) alculation of the line integal: Let the vecto function a have the following specification: a P,, + Q,, + R,, Let the paameteization of the cuve be defined b ( t) ( t) + ( t) + ( t) along the space () (4) i i i (5) diffeentiation of this equation ields: d t t i + t i + t i o dt d ( t) + + t i t i t i dt id + id + i d (6) fom which follows that i d d a a a,, Then the line integal () can be tansfomed to n a d lim a n Pn P,, i + P,, i + P,, i

31 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 99 a d P,, i P,, i P,, i d + + P,, i d P,, i d P,, i d + + P,, d + P,, d + P,, d a d P,, d + P,, d + P,, d this is the taditional fom without paentheses. Then appling di idt, a d P,, + P,, + P,, dt (7) b a Eample: Find a d fo the vecto function a p,q, p,q along the space cuve t cos t i + sin t i + ti t fom to π : Identif: t cos t ( t) ( t) sint ( t) t t sint t cos t Then, using equation (7), one ends up with a d π [ ] p sint + q cos t + t dt t p cos t + q sint + π π p+ p π p If the cuve is defined with the natual paameteization, then d d s ds T ds whee T is a unit vecto tangent to the cuve. dt ds dt dt Then d T ds. Recall also ( s) ds d + d + d + + dt The line integal then is calculated accoding to

32 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 a d b a P,, i T+ P,, i T+ P,, i T + + dt b at + + dt (8) a P,, i P,, i P,, i d + + Theefoe, the wo is pefomed onl b the tangential component of the foce. onsevative vecto fields If a is a gadient field of some scala field a ϕ (9) In this case function ϕ ( ) is called a potential function fo the gadient field a. Then a linea integal along the cuve connecting two points and is equal to the diffeence between values of the scala function at these end points: a d ϕ d dϕ ϕ ϕ a d ϕ d dϕ ϕ ϕ () It means that the same esult will occu fo an cuve connecting points and, and the line integal is said to be independent of path. We have fo a gadient field that dϕ ϕ d a d ( Pi Pi Pi) d ( Pi d Pi d Pi d ) Pd + Pd + Pd () Theefoe, Pd + Pd + Pd is an eact diffeential. o, the line integal P d + P d + P d is independent of path, if Pd + Pd + Pd dϕ is an eact diffeential. Test fo path independence Recall fom calculus that the diffeential fom Pd + Pd + Pdis an eact diffeential if and onl if P i P i () It is called the test fo path independence of a linea integal in space. Of paticula inteest ae the linea integals along the closed cuves denoted b iculation a d () It is called the ciculation of the vecto a aound the contou. If a is a foce field, then the ciculation is the wo done b the foce aound. It is obvious that fo the gadient field a d ϕ d If a vecto field is a gadient field of some scala field it is said to be consevative.

33 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7. OLUME INTEGRL onside a scala field ϕ ( ) and let be a volume. ubdivide the volume into subvolumes the volume as a limit and define an integal of the function ϕ ( ) ove ϕ ( ) d lim ϕ( ) (4) whee is an abita point in the subvolume. TRIPLE INTEGRL OF THE LR FIELD z One of the possible tpes of egion in which the volume integal can be educed to iteated tiple integal: z g (,),,z a a : f f g, z g, (,,z) Thee ae totall si standad tpes of volume egion which allow eduction to iteated integation: a a f (,) g (,) f ( ),,z a i a : f f i j i g, g, i j l i j i, j,l,, i j l The abita egion should be subdivided into combination of standad egions. onstuction of the iteated tiple integal of the scala field F (,,z ), (,,z) : F ( ) d F (,, z) dzdd (5) a f ( ) g (,) a f( ) g(,) If F, then the tiple integal ields the volume of the egion: d a f( ) g(,) dzdd (6) a f( ) g(,) Othe coodinate sstems (see Table 4): clindical: d ddθ dz, spheical: d sinθddφdθ.

34 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 Eample (Zill, p.54, modified) z z z onside the solid in the fist octant bounded b the the sufaces:, and z. Let the tempeatue distibution is given b T (,,z) + + z + 4 Find the aveage tempeatue of the solid T av : T av Td Fist, find the volume of the solid: T,, d dzdd dd ( ) d econd, integate tempeatue field ove the volume: Td + + z + 4 dzdd.77 integated with Maple Tempeatue distibution on the bottom of the solid Then the aveage tempeatue is: T,, T av Td.48 Rema: To find the volume, the poblem could be fomulated as a double integal Fd ( ) dd

35 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7. INTEGRL THEOREM GREEN THEOREM (Geen s Theoem Plane case) Let be a plane egion bounded b a closed contou. Let P(,) + Q(,) P(,) Q(,) whee P(, ),Q(, ),, a i j be a two-dimensional vecto field. Then ae continuous in simple closed cuve Q P Pd + Qd d (7) whee is taced in the diection such that egion appeas to the left of an obseve moving along contou. Equation (4) can be witten also in the following vecto fom: a d cula d (8) TOKE THEOREM (toes Theoem Geen s Theoem in pace) Let be a suface bounded b a closed contou. Let P(,,z) + Q(,,z) + R(,,z) whee Then a i j be a vecto field P P Q Q R R P,Q,R,,,,,, ae continuous in. z z R Q P R Q P Pd + Qd + Rdz cos (,) + cos (, ) + cos (,z) d z z n n n (9) n n Whee oientation of the suface is defined b the eteio unit nomal vecto n, and the contou is taced in the diection such that suface appeas to the left of an obseve moving along contou with the vecto n at points nea pointing fom the obseve s feet to his head. Equation (9) can be witten also in the following vecto fom: a d cula nd () claming that ciculation of a vecto field aound the bounda is equal to the flu of the cul though the suface.

36 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 4 DIERGENE THEOREM (the Gauss-Ostogads Theoem o the Divegence Theoem) Let be a volume bounded b a closed suface. a though the suface is equal to Then flu of the vecto field the integal of the divegence of the vecto field ove the volume a d div a d () Poof: We will show that equation () is appoimatel valid with an degee of accuac, i.e. that fo an ε >. a d - divad < ε ubdivide volume into such that a d - div a <δ This is possible accoding to the definition of the divegence as a limit (5). Multipl this inequalit b a d - div a <δ then summation ove all ields a a d - div <δ In this esult, the suface integal onl ove the eteio suface is left. ll inteio sufaces have to be the boundaies of some adjacent volumes and. Having the opposite nomal vectos, n n m, the suface integals ove cancel each othe in the summation: a d + a d m and m + an m d and and Let with, then accoding to the definition of the volume integal and lim diva ε hoose δ, then a d - divad <δ a d - divad < ε diva d fo an specified ε >. m

37 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 5 Recall Equations (85) and (): diva a a a + + a n a cos ni, + a cos ni, + a cos ni, a (nomal pojection) n Then the Divegence Theoem can be witten also in the following foms: a d divad an d divad () n a a d div d a a a a cos ( ni, ) + a cos ( ni, ) + a cos ( ni, ) d + + d The Divegence Theoem has a geat impotance in mathematical modeling in engineeing. In deivation of the govening equations fo phsical pocesses in the continuous media, the consevation laws ae applied to the contol volume ielding an equation which contains both suface integals and volume integals. pplication of the Divegence Theoem educes all integals to the volume integal which allows combination of all tems in one volume integal and concludes with the patial diffeential equation which govens the phsical pocess unde consideation.

38 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 6 EXMPLE (pplication of the Divegence Theoem - Equation of ontinuit) onside fluid flow with the velocit field v( ). The flu of the velocit field though some suface detemines the volume of fluid flowing though the suface pe unit time: q vn d Then fo incompessible fluid m ρq vn d is the mass of fluid flowing though the suface pe unit time. ρ ρ, so Fo compessible fluid, densit vaies in space m ρ vn d This equation epesents the net amount of fluid flowing though the suface with nomal vecto n epesenting the positive diection. Let now be the closed suface of some finite contol volume containing the point of space. In the stationa case, without an souces o sins, v n vn > m ρ vn d i.e. the mass of fluid flowing into is equal to the mass flowing out of. The mass of fluid in the contol volume is defined b the volume integal v ρd vn < n In the non-stationa fluid flow, densit depends on time (,t) ρ ρ Then the consevation of mass in the contol volume ields that the change of mass in the contol volume is equal to the mass flowing though the suface of the volume: ρd ρ d t vn (the sign is minus because the negative diection is towad the contol volume). Fo fied boundaies of contol volume, diffeentiation can be moved inside of the integal ρ d + ρ d t vn ppl the Divegence Theoem to eplace the suface integal b the volume integal ( ρv ) ρ d + div d t Now both integals ae ove the volume. dding the integals, we obtain

39 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 7 ρ + div( ρv ) d t Because this equation is valid fo an contol volume containing the point and the velocit and densit fields ae continuous, the integand should be equal to zeo povided that it also is continuous: equation of continuit ( ρ ) ρ + div v () t This equation is called the equation of continuit. Using opeato nabla, we can ewite it in the fom: ρ + ( ρv ) t ρ Fo incompessible fluid, densit does not depend on location, and t is negligible. Then the equation of continuit educes to v fo both stationa and non-stationa flow. Eecize onvet equation of continuit to atesian, clindical and spheical coodinates.

40 hapte I ecto and Tenso nalsis I. ecto nalsis - Revisited eptembe 5, 7 8 I.4 EXERIE REIEW QUETION: ) What is a vecto function of a scala vaiable? How it can be defined? What is the gaph of a vecto function? ) How can lines in space be defined? How can a segment in space be defined? ) What is the vecto equation defining a plane in space? What is the meaning of the constant tem in this equation? 4) What is the vecto equation of the plane containing thee fied points? 5) How is a sphee defined b a vecto equation? 6) What is the continuit of a vecto function? 7) How ae the vecto functions diffeentiated? 8) What is the geometical sense of the deivative of a vecto function? 9) What is the mechanical sense of the fist and second deivatives of a vecto function? ) Recall the main ules of diffeentiation of vecto functions. ) How is the Talo seies defined fo a vecto function? ) How ae the vecto functions integated? ) Give the geometical intepetation of the indefinite integal of a vecto function. 4) How is the definite integal defined fo a vecto function? 5) How can the length of a space cuve be calculated? 6) What is the paameteization of a cuve in tems of ac length? 7) What popet is possessed b the deivative of the space cuve paameteized in tems of ac length? 8) What is a scala field? 9) What is a vecto field? ) What ae level cuves and level sufaces? ) What is the opeato nabla? How is it applied to a scala field? ) What is a gadient of a scala field? Give a geometical intepetation of a gadient. ) What is a diectional deivative of a scala field? How it is defined with the help of opeato nabla? 4) What is the diectional deivative of a vecto field? How is it defined with the help of the opeato nabla? 5) How the suface integal of a vecto field calculated? 6) What is the flu of a vecto field though the suface? How is it defined? 7) What is the divegence of a vecto field? 8) What is the cul of the vecto field? 9) What is a line integal of a vecto field? ) What is a gadient field? ) What is it called when the line integal is independent of path? ) What does it mean that some gadient field is consevative? ) What is the ciculation of a vecto field? 4) Recall the Integal Theoems. 5) What is the sense of application of the Divegence Theoem fo deivation of the Equation of continuit and The Heat Equation?