= x. ˆ, eˆ. , eˆ. 5. Curvilinear Coordinates. See figures 2.11 and Cylindrical. Spherical

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1 Mathmatics Riw Polm Rholog 5. Cuilina Coodinats Clindical Sphical,,,,,, φ,, φ S figus 2. and 2.2 Ths coodinat sstms a otho-nomal, but th a not constant (th a with position). This causs som non-intuiti ffcts whn diatis a takn. Faith A. Moison, Michigan Tch U. Mathmatics Riw Polm Rholog 5. Cuilina Coodinats (continud) ( ) ( ) Fist, w nd to wit this in clindical coodinats. sol fo Catsian basis ctos and substitut abo cos sin sin cos cos sin substitut abo Faith A. Moison, Michigan Tch U.

2 2 Mathmatics Riw Faith A. Moison, Michigan Tch U. Polm Rholog 5. Cuilina Coodinats (continud) Rsult: Now, pocd: (W cannot us Einstin notation bcaus ths a not Catsian coodinats) Mathmatics Riw Faith A. Moison, Michigan Tch U. Polm Rholog 5. Cuilina Coodinats (continud) cos sin sin cos

3 3 Mathmatics Riw Faith A. Moison, Michigan Tch U. Polm Rholog 5. Cuilina Coodinats (continud) This tm is not intuiti, and appas bcaus th basis ctos in th cuilina coodinat sstms a with position.. Mathmatics Riw Faith A. Moison, Michigan Tch U. Polm Rholog 5. Cuilina Coodinats (continud) Final sult fo dignc of a cto in clindical coodinats:

4 Mathmatics Riw Polm Rholog 5. Cuilina Coodinats (continud) Cuilina Coodinats (summa) Th basis ctos a otho-nomal Th basis ctos a non-constant (a with position) Ths sstms a connint whn th flow sstm mimics th coodinat sufacs in cuilina coodinat sstms. W cannot us Einstin notation must us Tabls in Appndi C2 (pp ). Faith A. Moison, Michigan Tch U. Mathmatics Riw 6. Vcto and Tnso Thoms and initions Polm Rholog In Chapt 3 w iw Nwtonian fluid mchanics using th cto/tnso ocabula w ha land thus fa. W just nd a fw mo thoms to ppa us fo thos studis. Ths a psntd without poof. Gauss Dignc Thom V b dv S n b ds outwadl dictd unit nomal This thom stablishs th utilit of th dignc opation. Th intgal of th dignc of a cto fild o a olum is qual to th nt outwad flow of that popt though th bounding sufac. Faith A. Moison, Michigan Tch U. 4

5 Mathmatics Riw S ds n b Polm Rholog V Faith A. Moison, Michigan Tch U. Mathmatics Riw Polm Rholog 6. Vcto and Tnso Thoms (continud) Libnit Rul constant limits fo diffntiating intgals I di β α d d β α β α d f (, t d on dimnsion, constant limits Faith A. Moison, Michigan Tch U. 5

6 Mathmatics Riw Polm Rholog 6. Vcto and Tnso Thoms (continud) Libnit Rul fo diffntiating intgals J dj β ( α ( d d β ( α ( β ( α ( d aiabl limits f (, dβ dα d f ( β, f ( α, t on dimnsion, aiabl limits Faith A. Moison, Michigan Tch U. Mathmatics Riw Polm Rholog 6. Vcto and Tnso Thoms (continud) Libnit Rul fo diffntiating intgals J dj V ( d,, dv V ( V (,, dv f (,,, dv t S ( f ( n ) sufac ds th dimnsions, aiabl limits locit of th sufac lmnt ds Faith A. Moison, Michigan Tch U. 6

7 Mathmatics Riw Polm Rholog 6. Vcto and Tnso Thoms (continud) Substantial Diati Consid a function,, f tim at of chang of f along a chosn path f t t f d t d f f d t -componnt of locit along that path t d f f d t t d f t Whn th chosn path is th path of a fluid paticl, thn ths a th componnts of th paticl locitis. Faith A. Moison, Michigan Tch U. Mathmatics Riw Polm Rholog 6. Vcto and Tnso Thoms (continud) Substantial Diati Whn th chosn path is th path of a fluid paticl, thn th spac diatis a th componnts of th paticl locitis. along a paticl path f t f d f t t f f t d f f t t d f t f t along a paticl path Df Dt f t f Substantial Diati Faith A. Moison, Michigan Tch U. 7

E F. and H v. or A r and F r are dual of each other.

E F. and H v. or A r and F r are dual of each other. A Duality Thom: Consid th following quations as an xampl = A = F μ ε H A E A = jωa j ωμε A + β A = μ J μ A x y, z = J, y, z 4π E F ( A = jω F j ( F j β H F ωμε F + β F = ε M jβ ε F x, y, z = M, y, z 4π