By Synge, John Lighton; Schild, Alfred
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2. The conilgate tensor. Iowertng and raising strffires. From the covariant metric tensor onlvwe can obtain another 30 RrouarNralr Sracn tensor, also of the second order, but contravariant. 201. o : lo^nl: lazr azz lo*r';;; atn azN autt we shall suppose, here and throughout, that a is not znro. 202. 6tA. This follows from the ordinary rules for developing a determinant. 203. nlftt! L0 -- -. 204. 205. amra^' : 6i, Ar^Atm : d;. since a^oissymmetric, it is obvious that ant issymmetric also. 205, might be regarded as a ilefinition of a^', since either of these sets of equations determines these quantities uniquely.
A curve in Euclidean B-space has the equations fir: o @g tt, fi2: a sfura, f : bu, where *1, *, rf are rcctangular cartesian coordinates, u is a parameter, and o, b are positive constants. Find the length of this curye between the points u, :0 and u : 2r. what is the length of the curve with these same equations, between the same values of, u, if the metric form or trr" gspace is (d*), +(it#)o - @*'1r7 Consider your result for the caseswhere o is greater than, equal to, and lessthan D. unless we are dealing with a null curye, for which s : 0, we may take s as parameter along the curve.
W)- 0w a*-,' We ask: Is/, a tensor? 432. vector, and so dx| f'- 0"#. 433. 434. 0w 0w 0p' 0w 0x' -: ilptr Ap, AO', Ap"Adr' and d/aw\ - d / a w \ d * . #,:#,#+##. 432, with a changeof suffixes. 437. -( -. - - dr'\ ,\/" f"a*,,) - 0w 02*' dd^ 0P" d*t^O*t' du Hencef, is o cnar,iant uedor. 0u -5tw aP" 02*" ax"a*" : e. 4is a formal processwhich holds equally well when an arbitrary parameteru is employedinstead of s. Hence d2* d,cc^dr' fr= ar^ d", * lmn, ,l do do is a covariant vector, ffid d2*' ( rldx^dx" f : dur+\*"| * * is a contravariant vector.