By Ernest W. Flick
This publication (Volume 2) provides a number of hundred complex cleansing product formulations for loved ones, business and car purposes. All formulations are different from these in different volumes, so there isn't any repetition among volumes.
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Extra resources for Advanced Cleaning Product Formulations, Volume 2
Let P be a point and Q be a nearby point of a space curve x(s). Let ∆ϕ be the angle between the tangents in P and Q and let ∆ψ be the angle between the osculation planes (binormals) in P and Q. Then the curvature ρ and the torsion τ in P are defined by: dϕ ds ρ2 = 2 = lim ∆s→0 2 ∆ϕ ∆s , τ2 = dψ ds 2 and ρ > 0. For plane curves ρ is the ordinary curvature and τ = 0. The following holds: ¨ x ¨) ρ2 = ( , ) = (x, ˙ ˙ and τ 2 = (b, b) Frenet’s equations express the derivatives as linear combinations of these vectors: ˙ ˙ = ρn , n˙ = −ρ + τ b , b = −τ n ...
6 27 Properties of Bessel functions Bessel functions are orthogonal with respect to the weight function p(x) = x. J−n (x) = (−1)n Jn (x). The Neumann functions Nm (x) are definied as: Nm (x) = 1 1 Jm (x) ln(x) + m 2π x ∞ αn x2n n=0 The following holds: lim Jm (x) = xm , lim Nm (x) = x−m for m = 0, lim N0 (x) = ln(x). x→0 lim H(r) = r→∞ x→0 e±ikr eiωt √ , r lim Jn (x) = x→∞ x→0 2 cos(x − xn ) , πx lim J−n (x) = x→∞ 2 sin(x − xn ) πx with xn = 12 π(n + 12 ). 8 ex dn (−1)m n m n −x x e = x n! dxn m!
Special cases are n = 0, 1, 2. 1 Some special cases Frobenius’ method Given the LDE d2 y(x) b(x) dy(x) c(x) + + 2 y(x) = 0 dx2 x dx x Chapter 4: Differential equations 25 with b(x) and c(x) analytical at x = 0. This LDE has at least one solution of the form ∞ ri an xn yi (x) = x with i = 1, 2 n=0 with r real or complex and chosen so that a0 = 0. When one expands b(x) and c(x) as b(x) = b0 + b1 x + b2 x2 + ... , it follows for r: r2 + (b0 − 1)r + c0 = 0 There are now 3 possibilities: 1. r1 = r2 : than y(x) = y1 (x) ln |x| + y2 (x).
Advanced Cleaning Product Formulations, Volume 2 by Ernest W. Flick