A First Course in the Differential And Integral Calculus

Cover A First Course in the Differential And Integral Calculus
A First Course in the Differential And Integral Calculus
William F William Fogg Osgood
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Hence (2) k=±—, or better: K = \ — w ds Ids From the foregoing definition we see that the curvature is the rate at which the tangent turns when a point describes the curve wifh unit velocity.
To compute dr/ds we have (3) tanr = ^ or r = tan" 1 ^.
dx dx It will be convenient to introduce a shorter notation for deriva- tives and we shall adopt Lagrange's, which employs accents : y *=/(?), It follows, then, that dy' = ( &dx = ( ^ ) dx = y"dx dx dx 2 and ds = Vdx 2 -f dy 2 = VI -\-y' 2 dx.
Returnin
...g to (3) and differentiating we have: 136 CALCULUS r-taa-V, ^=«^ = fJE dr ds i+y' 2 i+y' 2) . '/ (l+y, 2 P (4) I y" I (1+2/' 2 ) [ ♦*r The reciprocal of the curvature is called the radius of curva- ture and is usually denoted by p : * l__ (l+y") f L dtf \y"\ " (?) /> = K = dx 2 ; The radius of curvature of a circle is its radius. The curva- ture of a curve at a point of inflection is in general ; for y" = at such a point if y" is continuous there.
Example. To find the curvature of the parabola f 2mx.


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