By Hermann Bondi
ISBN-10: 0486240215
ISBN-13: 9780486240213
Greatly reoriented presentation of Einstein's certain thought and considered one of Most worthy well known bills to be had derives relativity from Newtonian rules, instead of against them. 60 illustrations.
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Sample text
Now recalling the function of the reciprocal base vectors we have e (i )k e( j )k = δji . 44) whence we conclude that the bracketed term on the left satisfies e(i ) e (i )j = δ . 45) That is, the sum over the base vector label yields the Kronecker delta in the component indices. 45)). 24), and use Cartesian components for each vector. 29) as g m = e( ) · e(m) . 2 Let us recall that by the existence of the inverse metric matrix g i gij = δj . 48) g i e(ik ) e( j )k = δj . 45) to obtain ( j) g ( )i e(im) = e ( )m .
4). 4) in the form ei · ej = δji . 21) These vectors are called reciprocal base vectors for fairly obvious reasons. The symbol δji is again the Kronecker delta, but the up and down indices remind us of the necessity of a set of reciprocal base vectors. The reciprocal base vector may be thought of as a row vector (a 1 × 3 matrix, rather than a 3 × 1 matrix for a column vector) that is matrix multiplied into the column base vector to effect a ‘dot’, that is a scalar, product. One practical definition of these reciprocal base vectors is (we use Greek letters to indicate that no sums are intended when they are repeated in a product) eα ≡ eβ ∧ e γ .
10 Find the complete Euler angle matrix that gives the primed base vectors in terms of the Cartesian base vectors by calculating S ≡ S S S . 11 Show that for each of the individual Euler angle rotation matrices of the preceding question Sα Sα = 1 where Sα is the transpose of Sα and 1 is the unit 3 × 3 matrix. Here α is any of φ, θ or ψ. Note that this property of ‘orthogonality’ also applies to S by using the transpose of a matrix product. 5 The figure shows a Cartesian reference frame {X, Y, Z}, together with axes {ζ , η, Z} that are rotated through the angle φ about the Z axis.