By Thomas R. Kane

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**Sample text**

Result: See Fig. 3b. FIG. 5) of v at z* + h and at 2*, where z* is a particular value of z and h a scalar having the same dimensions as 2, can be expressed in terms of values of derivatives of v with respect to z in R at z*, as follows : R h Rdv\ h? RcN\ *+A "~ V U* 1! dz\ 2* 2 ! dz2 + 1, 2, 3, be unit vectors (not parallel to the Proof: Let nt, i same plane) fixed in R. 3) and, by Taylor's theorem for scalar functions, WdhJ , « _ h^dvj\ v i\z*+h ~~ ΌΑζ* ~ Wfa 2! dz2 + +. The values of v at z* and at z* + h, expressed in terms of their nt-, i = 1, 2, 3, components, are 3 3 1=1 Diff.

2)1/2 = (A2P'2 + hV ' p" + U/2 p' + hhp" + ■ ■ ■ (p'2 + hp' · p" + Substitute : p' + *ftp" + . . SS (ρ'2 + Ap' - p" + . 1, suppose that there exists a point 0 which is fixed in both R and R' and that p is the position vector of a point P relative to 0. Then P traces out a curve C in R and curve C in R'. Determine the cosine of the angle a between the tangents to C and C" at point P, for t = 0. Solution: Let r and τ' be vector tangents of C and C at P . 1, R 'dp\ = 3n2 + 4n! dt i = 0 SECS. 3 Diß.

Of Vectors] SECS. 1) ds/ Hence τ = R 33 dp\ ds\ és. ds While this expression for r appears to be simpler than the one given in Sec. 1, it is frequently less convenient, because the functional dependence of p on s is often more complicated than p's dependence on some other variable. 4 The plane passing through P and normal to τ is called the normal plane of C at P. 1 If a curve C is fixed in a reference frame R (see Fig. 1a) and B is the binormal to C at a point P of C, a unit vector ß parallel to B is called a vector binormal of C at P and is given by ß = ρ' X Ρ' IP' X Ρ'Ί where p is the position vector of P relative to a point 0 fixed in R and primes denote differentiation with respect to z in R, z being any scalar variable such that the position of P on C depends on z.