Surfaces in the space E
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1 3 Srfaces in he sace E
2 Le ecor fncion in wo ariables be efine on region R = x y y whose scalar coorinae fncions x y y are a leas once iffereniable on region. Hoograh of ecor fncion is a iece-wise smooh srface in E 3. For any orere air of real nmbers he fncion ale is osiion ecor = x y y of srface oin P wih Caresian coorinaes Orere air of nmbers oin on srface. P = [x y z ]. is calle crilinear coorinaes of If R is a reglar region wih bonary in a close reglar cre in E hen srface is calle elemenary srface or srface ach.
3 All oins on srface eermine by crilinear coorinaes = cons. res. = cons. are oin on one cre on he srface which is calle iso-arameric -cre - = x y z or iso-arameric -cre - = x y z. Iso-arameric cres form wo sysems of cres on srface hey form crilinear ne of cres on srface calle iso-arameric ne of cres. Any cre from one sysem of cres inersecs all cres in he oher sysem of cres on he srface. One iso-arameric cre from each of he wo sysems is assing hrogh any srface oin while i hols P =
4 Poin P on he srface is calle reglar oin if here exis coninos firs arial eriaies of ecor fncion on some neighborhoo of an i hols In he oosie case when ecors are linearly eenen we seak abo singlar oin on srface. Vecor fncions z y x z y x efine irecion ecor fiels of angens o iso-arameric cres on srface assing hrogh he srface reglar oin P.
5 A any reglar oin P here exis linearly ineenen ecors angen ecor o iso-arameric -cre an angen ecor o iso-arameric -cre z y x z y x ha eermine a niqe angen lane a he gien oin P Eqaion of angen lane ] [ P z y x z y x z Z y Y x X
6 Vecor n is calle normal ecor o srface a he reglar oin P. Line eermine by irecion ecor n an assing hrogh oin P is a normal line o srface ha is ereniclar o he angen lane o srface in his oin. n Vecor fncion efines irecion ecor fiel of normals o srface in reglar oins an eermines orienaion of srface. Vale of fncion n is a ni ecor whose coorinaes are irecion cosines of normals o srface a he reglar oins while n cos i cos j cos where are angles ha ecor n forms o ni ecors i j k. k
7 Le I is cre on srface R. Tangen ecor o he cre can be reresene by ifferenial Sqare of his ifferenial saisfies eqaion Denoing E F G we can receie form E + F + G ha is calle he firs fnamenal ifferenial form of srface or he firs ensor of srface enoe also as. The firs ifferenial form of srface escribes inerior roeries of srface geomery lenghs of cre segmens an heir angles. Formla D = EG - F is calle iscriminan of he firs ifferenial form.
8 The firs ifferenial form of srface is osiiely efinie a any reglar oin on srface an i hols:. E =. EG - F = G = - = 3. symmeric means ha scalar roc is commaie Lengh of cre segmen on srface Le a b is a cre on srface R whose firs ifferenial form is s = E + F + G For he lengh sa B of segmen where A = a a B = b b hols s A B b a s b a E F G
9 Angle of wo cres on srface Le cres k an k be eermine by heir arameerizaions an a b on srface R an le hey inersec in he srface oin P whose crilinear coorinae on he cre k is an on he cre k i is. Angle of hese cres can be calclae as angle of heir angen ecors an he following hols.. cos G F E G F E G F E For he angle of iso-arameric cres a he oin P i hols EG F.. cos
10 Area of elemenary srface ach Area elemen on a srface efine by ecor fncion ha is a leas once iffereniable on region R is arallelogram in he angen lane o srface eermine by ecors an forming angle F EG S. sin. Area of srface ach efine on region can be calclae by means of oble inegral from he area elemen on he srface D S
11 The secon fnamenal ifferenial form of a srface or he secon ensor of srface is relae o he srface crare in he reglar oin an i is enoe while for is coefficiens hols L M N = n. = L + M + N n. n. n.... Exression D = LN M is calle iscriminan of he secon ifferenial form. [ [ [ D D D ] ] ]
12 Nmber LN EG is calle oal or Gassian crare in he gien oin. K Srface oin a which K is calle srface elliic oin. Tangen lane a his oin has js one oin in common wih he srface while srface is locae in one half-sace eermine by angen lane an normal ecor o he srface a his oin. Srface oin a which K is calle srface hyerbolic oin. Tangen lane a his oin inersecs srface in a srface cre which has a oble oin a he angen oin while srface is locae in boh half-saces eermine by angen lane a his oin. Srface oin a which K = is calle srface arabolic oin. Tangen lane a his oin is angen o he srface in a cre while srface is locae in one half-sace eermine by angen lane an normal ecor o he srface a his oin. M F D D
13 Theorema egregim Gassian crare of he srface can be eermine by means of coefficiens of he firs fnamenal ifferenial form of srface an heir eriaies. Srfaces wih elliic oins shere elisoi araboloi Srfaces wih arabolic oins lane cylinrical srface conical srface Srfaces wih hyerbolic oins hyerboloi seoshere
14 Srface wih all yes of oins ors
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