. Applied calculus; principles and applications . ogarithmic curve y = 6*. 7. The catenary i/ = ^ (e^/«+ e^/^). 8. The hypocycloid x + 2/ = a^. 9. The curve x^ -\-y^ = aK 93. Radius of Curvature in Polar Coordinates. — From (4) and (10) of Art. 67, ^ = 6 + ^, ^ = tan-i^; •*• dd~ ^ dd dd p2 _|_ (^dp/doy d4> ^p^-^2 {dp/dey - p. (IV/c^6>^, •• de p +{dp/dey . ^ ds/dl ^ [p^ + (^p/(i^)^]^ ,. 77 .ON •• ^ dct>/dd p^ + 2(dp/ddy-pdp/dd ^^^-^^^^ Corollary. ■.— Since R = cc at a point of inflexion, p^ + 2{dp/dey-p.^,= o is a necessary condition for such a point. K = d4>^ ds 6aV (9 y + a)^ dcj)
. Applied calculus; principles and applications . ogarithmic curve y = 6*. 7. The catenary i/ = ^ (e^/«+ e^/^). 8. The hypocycloid x + 2/ = a^. 9. The curve x^ -\-y^ = aK 93. Radius of Curvature in Polar Coordinates. — From (4) and (10) of Art. 67, ^ = 6 + ^, ^ = tan-i^; •*• dd~ ^ dd dd p2 _|_ (^dp/doy d4> ^p^-^2 {dp/dey - p. (IV/c^6>^, •• de p +{dp/dey . ^ ds/dl ^ [p^ + (^p/(i^)^]^ ,. 77 .ON •• ^ dct>/dd p^ + 2(dp/ddy-pdp/dd ^^^-^^^^ Corollary. ■.— Since R = cc at a point of inflexion, p^ + 2{dp/dey-p.^,= o is a necessary condition for such a point. K = d4>^ ds 6aV (9 y + a)^ dcj) my ds {m + y .)i dds a R = - 3 {axy)K K = d(i> a^ 2( x + y)^ RADIUS OF CURVATURE IN POLAR COORDINATES 141 Example. — To find R for the curve p = sin ^. Heredpdd R = (p^ + cosH)i (sin2^ + cos2 6i)^ p2+2cos2^-p(-sin6) sin2 0+2cos2^+sin2^ 2 This curve p = sin 6, a circle with ^ unit diameter, in connection with theformula for polar curves, tan \l/ = ;, furnishes a derivation of the dp/dd d (sin 6). Since for circle. \l^ = d, tan d dp/dd dp p dd sin 0 tan 6 tan 0 that is, d (sin 6) = cos 6 from figure: OP p dd = cos ddd; tSLlid = OA cos (9 and since tan^ dpdd dp/dd = cos d = subnormal OA; so again, d (sin d) = cos d dd. This curve serves as an illustration of maxima and minimain polar coordinates. Thus, p = sin 0 will be a maximum or a minimum when -^ = cos 0 = 0, when ^ = - or -^ ; and since -^^ = — sin d, is negative when ^ = o > . TT ^ . , ., d^p • /. • p = sm - = 1 IS a maximum, while -^ = — sm 0 is posi- 142 DIFFERENTIAL CALCULUS — 1 is a minimum. trv€ when 6 = -x-, .*. p = sm 2 • • ^ - 2As the denominator of the fractional value of E is 2 for anyvalue of d, there is no inflexion point, R not being infiniteat any point. EXERCISE Xni. Find R in each of the following curves: 1. The Cardioid p = a (1 — cos0). 2. The Lemniscate p^ = a^ cos 2 is an inflexion point ? 3. The Spiral of Archimedes p = ad. R = 2V2ap/3. + 2 4.
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