Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORS26.28 A curve r(t) is parameterised by a scalar variable t. Show that the length of thecurve between two points, A and B, isgivenby∫ √ BduL = g i du jijA dt dt dt.Using the calculus of variations (see chapter 22), show that the curve r(t) thatminimises L satisfies the equationd 2 u idt 2du j du k+Γi jk = ¨ṡ du idt dt s dt ,where s is the arc length along the curve, ṡ = ds/dt and ¨s = d 2 s/dt 2 . Hence, showthat if the parameter t is of the form t = as + b, wherea and b are constants,then we recover the equation for a geodesic (26.101).[ A parameter which, like t, is the sum of a linear transformation of s and atranslation is called an affine parameter. ]26.29 We may define Christoffel symbols of the first kind byΓ ijk = g il Γ l jk.Show that these are given byΓ kij = 1 2By permuting indices, verify that(∂gik∂u j+ ∂g jk∂u i− ∂g ij∂u k ).∂g ij∂u =Γ k ijk +Γ jik .Using the fact that Γ l jk =Γl kj, show thatg ij; k ≡ 0,i.e. that the covariant derivative of the metric tensor is identically zero in allcoordinate systems.26.24 Hints and answers26.1 (a) u ′ 1 = x 1 cos(φ − θ) − x 2 sin(φ − θ), etc.;(b) u ′ 11 = s2 x 2 1 − 2scx 1x 2 + c 2 x 2 2 ≠ c2 x 2 2 + csx 1x 2 + scx 1 x 2 + s 2 x 2 1 .26.3 (a) (1/ √ 2)( √ 2, 0, 0; 0, 1, −1; 0, 1, 1). (b) (1/ √ 2)(1, 0, −1; 0, √ 2, 0; 1, 0, 1).r =(2 √ 2, −1+ √ 2, −1 − √ 2) T .26.5 Twice contract the array with the outer product of (x, y, z) with itself, thusobtaining the expression −(x 2 + y 2 + z 2 ) 2 , which is an invariant and therefore ascalar.26.7 Write A j (∂A i /∂x j )as∂(A i A j )/∂x j − A i (∂A j /∂x j ).26.9 (i) Write out the expression for |A T |, contract both sides of the equation with ɛ lmnand pick out the expression for |A| ontheRHS.Notethatɛ lmn ɛ lmn is a numericalscalar.(iii) Each non-zero term on the RHS contains any particular row index once andonly once. The same can be said for the Levi–Civita symbol on the LHS. Thusinterchanging two rows is equivalent to interchanging two of the subscripts ofɛ lmn , and thereby reversing its sign. Consequently, the magnitude of |A| remainsthe same but its sign is changed.(v) If, say, A pi = λA pj , for some particular pair of values i and j and all p then,982