Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

102 Extrema and the Geometric Applications of a Derivative [Ch. 3
Denoting by a the angle formed by the tangent (in the direction of
increasing arc of the curve s) with the positive ^-direction, we get
In polar coordinates,
dx
cos a = -3- ,
ds
dy
sina -r .
ds
Denoting by p the angle between the radius vector of the point of the
curve and the tangent to the curve at this point, we have
a
dr
008 P = '
sin p
/
2. Curvature of a curve. The curvature K of a curve at one of its
points M is the limit of the ratio of the angle between the positive directions
of the tangents at the points M and N of the curve (angle of contin-
gence) to the length of the arc ^MN^\s when .V M (Fig. 35), that is,
K= iim Au = ^, *
A s * o A S rfs
\\hore a is the angle between the positive directions of the tangent
point M arid the .v-axis.
jt the
The radius of curvature R is the reciprocal of the absolute value of the
curvature,
i. e.,
The circle f K = ,
where a is the radius of the circle) and the straight
line (/C = 0) are lines of constant curvature.