Integral Vector Operations and Related Theorems Applications in Mechanics and E&M

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1 Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts Z X Y A cuve as shown can be epesented by ( = x( y( j z( k = xn ( n whee x =x, x =y, and x =z.,, ae espectvely, j, and k. Ths paametc epesentaton utlzes the paamete t. Let a t b. The dffeental dsplacement vecto along s d = dx dyj dzk = dx n n The followng ntegals along, whee may be closed o open, ae called lne ntegals (they nvolve d ): φd f d (a) (b) (c) f d We wll efe to (b) as a scala lne ntegal (the ntegand, a dot poduct, s eally not a vecto). If cuve s closed, the lne ntegal ae noted as: φ d f d (a) (b) (c) f d o whee the aows ndcates the decton of the loopng along. The scala lne ntegal of f along a closed cuve s called the cculaton of f : cul( f ) = f d mlaly, a suface can be epesented by ( = x( y( j z( k whee u and v Z ae paametes. The dffeental suface element s: X Y

2 Dola Bagayoko (0) ) = dudv = n whee nˆ s the unt vecto nomal (pependcula) to the suface element. u v = n ˆ uface ntegals ae ntegals whee the dffeental element s s d : φ f f The scala suface ntegal f s called the flux of f though the suface aea. If the suface s closed, lke the sphee above, the suface ntegals ae noted as : φ f and f Let dv = dxdydz be the volume element n a thee dmensonal (-D) ectangula coodnate system as the one shown above, then φ dv and dv =, y = f, whee φ = φ( x, y, z) and f = f ( x, y, z) ae called volume ntegals. Fo φ =, dv = v, the volume enclosed between the planes defned by x =, x =, y, and z =, z = ONNETIVITY OF DOMAIN: A two dmensonal (-D) doman o egon s sad to be smply connected f and only f the egon wthn any closed cuve that s entely n the doman s also (the egon s) entely n the doman. mply connected not smply connected. A -D doman s smply connected f and only f the egon enclosed by any closed suface n the doman s also entely n the doman. The essental dffeence between lne and suface ntegals and the odnay (defnte o ndefnte) ntegal wth one vaable of pe-calculus s llustated below. Poblem: Let φ = xy a) uve whee, get φd y = x and z = 0 fom () to (,,0) along and

3 Dola Bagayoko (0) b) uve epesentng the staght lne x = t, y = t, z = 0 fom () to (,,0). olutons a) uve s a paabola. It can be paametcally epesented by x = t and y = t φ = ( t )( t ) = t (ecall that z = 0). ubsttutng these values and d = dx dyj dzk, the ntegal s,,0,,0 t ( dx dyj dzk ) = t ( dt tdtj 0k ) j = 4 5 b) Fo x = t and y = t wth z = o, d = dt dtj and φ = ( t )( = t. Hence, the lne ntegal s t dt 0,, 0 φ d = =,,0 t ( j) dt j = t dt j 0 lealy, the value of the lne ntegal, between two fxed ponts, depen on the cuve followed fom one pont to the othe. Mechancs the wok done by a foce F, n movng fom one pont a to anothe b geneally depen on the path (cuve) followed fom a to b. b b W = F d = a (δw) depen on the path. a onsevatve Foces : F b s consevatve f F d s ndependent of the path fom a to be. In a that case, F. d = 0! A consevatve foce can always be wtten as the gadent of some scala functon: F = V whee V s a scala potental enegy functon. II. Physcal Meanngs of φ = ˆ φ, E, A φ φ φ = j k = Lm = volume of the egon of space bounded by the closed suface. φ d = dx φ φ. dy j. j dzk. k = dφ,. = k. k = j. j = *The value of φ s equal n magntude to the maxmum dectonal devatve of φ : φ max s = φ *The decton of φ, at a pont, s that of maxmum vaaton of φ at that pont. E E E x y Ez E v = = E = Lm E v 0 s the volume bounded by the closed suface. The pont whee E s evaluated s nsde the volume V. v 0 φ

4 Dola Bagayoko (0) E s the net flux (outflow o nflow) of E though the closed suface. E v s a measue of souce () o snk (-) stength at the pont whee t s evaluated. A ) = A jk J ˆ k A = nmax Lm x A d = Lm A 0 0 A s a measue of the ntensty of cculaton aound the pont whee A s evaluated. The decton of A at a pont s that fo maxmum cculaton. NOTE: The above denttes, whch expound on the meanngs of, F (o E v ), and F (o A ), dectly lead to the GADIENT, DIVEGENT (o Gauss), and UL THEOEM that follow. TO KNOW (Thee fomula n one, dependng on *) αdv = α (,, ) The thee theoems ae summazed n ths sngle expesson. When * s blank, t s the gadent theoem; when * s a dot, t s the dvegence o Gauss theoem; and when the * s a coss, then we have the cul theoem. GADIENT THEOEM (When the sta above s eplaced wth blank): dv = φ Volume enclosed by uface (closed) φ = cala functon contnuous n. Note well that ths theoem and = Lm φ Lead dectly one to the othe 0 V DIVEGENE (O GAU) THEOEM (When the sta above s eplaced wth a do: Volume enclosed by suface. FdV = F F = a vecto functon contnuous (eveywhee) n. Note also that ths theoem and F = Lm F lead dectly one to the othe. 0 UL THEOEM (When the sta above s eplaced wth a coss): FdV = F Volume enclosed by suface. F = a vecto functon contnuous n Agan, ths theoem and F = Lm F lead dectly one to the othe. 0 These theoems actually epesent tansfomatons. They tansfom the volume ntegal -- of a knd of devatve of a functon -- to a suface ntegal of the functon. GEEN THEOEM (φ Ψ ψ ) dv = φ ψ 4

5 Dola Bagayoko (0) * egon bounded by the closed suface * φ and Ψ have contnuous second devatves. ( φ Ψ ψ φ) dv = ( φ ψ ψ) * Volume enclosed by uface * φ and Ψ have contnuous fst devatves. TOKE THEOEM AND ELATED ONE ( F ) = F d s the suface bounded by the closed smple cuve. F s a vecto functon wth contnuous fst patal devatves. ecall that del o nabla, ~ ˆ = j k x, can be teated as a vecto. Note, howeve, that t has no meanng f somethng to dffeentate does not exst on ts ght. Usng the popety of the tple poduct, ( A B) = A ( B ), we ewte the ntegand above as ( F) = ( F) = ( ) F Hence, the tokes theoem can be also be wtten as ( ) F = d F. elated Theoems = φd ( ) F = All these thee theoems can be educed to: d F ( ) α = d α uface bounded by close cuve. α s a scala o vecto functon wth contnuous devatves If α =φ, a scala functon, then cannot be o -- hence, t s blank. If α = F, a vecto functon, then cannot be blank. Hence t s (fo the dot poduc o (fo coss poduc. The TOKE and elated theoems tansfom suface ntegals nto (closed) lne ntegal and vcevesa. The gadent, dvegence, and cul theoems tansfom volume ntegals nto (closed) suface ntegals and vce-vesa. Please note the condtons of contnuty of functons o of the devatves; wheneve noted, these condtons ae necessay fo these theoems to hold. 5