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

Hence, on the basis of formulas (1) and (2), we get
and
1
R ~( fl i + 6i)
JDL VaT+b*^ a
a
/. "fl' + a 1
1 a*b = b
~ 2
a
Q
(a 2 + b*)~~ a 2 '
2 + b
Thus, for a screw-line, the curvature and torsion are constants.
3 Frenet formulas:
dr _ v ^v ___ T iP ^P_ v
7s~"~R '
5s ~~
'
~~~~R~T~~Q ds~~~~o"'
2104. Prove that if the curvature at all points of a line is
zero, then the line is a straight line.
2105. Prove that if the torsion at all points of a curve is zero,
then the
2106.
curve is a
Prove that
plane curve.
the curve
x=\+3t + 2t 2
, y = 22t + 5t\ z=lt*
is a plane curve; find the plane in which it lies.
2107. Compute the curvature of the following curves:
a) x = cost, y = s'mt, z = cosh / at the / point = 0;
b) x* 2
// -| z 2 2 = l, y -2x + z = Q at the point (1, 1, 1).
2108. Compute the curvature and torsion at curves:
any point of the
a) je = e'cos/, y = e i
sint, z e*\
b) x^acosht, y asiuht. z = at (hyperbolic screw-line).
2109. Find the radii of curvature and torsion at an arbitrary
point (x, y, z) of the curves:
a) x 2 =
b) x 9 =
2110. Prove that the tangential and normal components of
acceleration w are expressed by the formulas
dv v 2
VOif T