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By Gray A., Mathews G.B.

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Extra resources for A treatise on Bessel functions and their applications to physics

Sample text

1. Find s, and find the parametric description of the curve given by the arc length. It is proved in Differential Geometry that any plane curve has a curvature κ(t) = {ez × r (t)} · r (t) , r (t) 3 where we let the plane of the curve be the (X, Y )-plane in the space. 2. Prove that κ is proportional to s for K. The curve under consideration has many names: the clothoid, Euler’s spiral, Cornu’s spiral. Remark. e. Ancient Greek: κλωθω = I spin. ♦ A Parametric description with respect to arc length, curvature.

1) Alternatively, 2 + 2 d dϕ = √ ϕ 2e , hence 1 K 4) Here ds = 0 √ √ ϕ 2 e dϕ = 2(e − 1). ϕ ϕ d = a · sin2 · cos , so dϕ 3 3 2 + d dϕ 2 =a sin6 ϕ ϕ ϕ ϕ + sin4 · cos2 = a · sin2 , 3 3 3 3 thus 3π K ds = 0 a sin2 ϕ dϕ = 3a 3 π 0 sin2 t dt = 3a 2 π 0 (1 − cos 2t) dt = 3aπ . 1. 3 Below are given some space curves by their parametric descriptions x = r(t), t ∈ I. Express for each of the curves there parametric description with respect to arc length from the point of the parametric value t0 . 1) The curve r(t) = (cos t, sin t, ln cos t), from t0 = 0 in the interval I = 0, π .

A Parametric description by the arc length. D Find s (t) = r (t) and then s = s(t) and t = τ (s), where we integrate from t0 . Finally, insert in x = r(t) = r(τ (s)). com 39 Calculus 2c-7 Arc lengths and parametric descriptions by the arc length hence t s(t) = 0 1 ln 2 1 + sin u 1 − sin u t 0 cos u du = 1 − sin2 u t = 0 1 ln 2 t 0 1 + sin t 1 − sin t 1 1 + 1 + sin u 1 − sin u 1 2 cos u du . 2. Then 1 + sin t = e2s , 1 − sin t dvs. Notice that it follows from t ∈ 0, cos t = e2s − 1 = tanh s, e2s + 1 sin t = s ≥ 0.

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A treatise on Bessel functions and their applications to physics by Gray A., Mathews G.B.


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