Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.20 Vector operators in tensor form where the expression in parentheses is the required covariant derivative Tij;k= ∂Tij ∂uk + ...

TENSORS In order to compare the results obtained here with those given in section 10.10 for orthogonal coordinates, it is necess ...

26.20 VECTOR OPERATORS IN TENSOR FORM
SupposeA=[aij],B=[bij]and thatB=A−^1. By considering the determinanta=|A|, show that ∂a ∂ ...

TENSORS and so the covariant components ofvare given byvi=∂φ/∂ui. In (26.97), however, we require the contravariant componentsvi ...

26.21 ABSOLUTE DERIVATIVES ALONG CURVES this is the analogue of the expression in Cartesian coordinates discussed in section 26. ...

TENSORS components of a second-order tensorTare δTij δt ≡T ij ;k duk dt , δTij δt ≡Tij;k duk dt , δTij δt ≡Tij;k duk dt . The de ...

26.23 EXERCISES Writing out the covariant derivative, we obtain ( dti ds +Γijktj duk ds ) ei= 0. But, sincetj=duj/ds, it follows ...

TENSORS 26.3 In section 26.3 the transformation matrix for a rotation of the coordinate axes was derived, and this approach is u ...

26.23 EXERCISES 26.10 A symmetric second-order Cartesian tensor is defined by Tij=δij− 3 xixj. Evaluate the following surface in ...

TENSORS (b) Find the principal axes and verify that they are orthogonal. 26.17 A rigid body consists of eight particles, each of ...

26.23 EXERCISES 26.23 A fourth-order tensorTijklhas the properties Tjikl=−Tijkl,Tijlk=−Tijkl. Prove that for any such tensor the ...

TENSORS 26.28 A curver(t) is parameterised by a scalar variablet. Show that the length of the curve between two points,AandB,isg ...

26.24 HINTS AND ANSWERS in the (multiple) summation on the RHS, eachAnkappears multiplied by (with no summation overiandj) ijkA ...

27 Numerical methods It happens frequently that the end product of a calculation or piece of analysis is one or more algebraic o ...

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS errors introduced as a result of approximations made in setting up the numerical pro ...

NUMERICAL METHODS 0. 20. 4 0. 6 0. 81. 0 1. 2 1. 4 1. 6 1. 8 − 4 − 2 0 2 4 6 8 10 12 14 x f(x) f(x)=x^5 − 2 x^2 − 3 Figure 27.1 ...

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS root byξand the values of successive approximations byx 1 ,x 2 ,...,xn,.... Then, fo ...

NUMERICAL METHODS nxn f(xn) 1 1.7 5.42 2 1.544 18 1.01 3 1.506 86 2. 28 × 10 −^1 4 1.497 92 5. 37 × 10 −^2 5 1.495 78 1. 28 × 10 ...

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 1. 0 1. 0 1. 0 1. 0 1. 2 1. 2 1. 2 1. 2 1. 4 1. 4 1. 4 1. 4 1. 6 1. 6 1. 6 1. 6 − 4 ...

NUMERICAL METHODS nAn f(An) Bn f(Bn) xn f(xn) 1 1.0000 − 4 .0000 1.7000 5.4186 1.3500 − 2. 1610 2 1.3500 − 2 .1610 1.7000 5.4186 ...