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

234 Functions of Several Variables [Ch. 6
2069. Investigate the character of the "discriminant curves"
of families of the following lines (C is a constant parameter):
a) cubic 8
parabolas y = (x C) ;
2
b) semicubical parabolas = t/ (x C)*;
c) Neile 2
parabolas y* = (x C) ;
d) strophoids (a + x) (y C) 8 =* 2
(a x).
Fig. 83
2070. The equation of the trajectory of a shell fired from a
point with initial velocity v at an angle a to the horizon
(air resistance disregarded) is
Taking the angle a as the parameter, find the envelope of all
trajectories bf the shell located in one and the same vertical
plane ("safety parabola' 1
) (Fig. 83).
Sec. 17. Arc Length of a Space Curve
The differential of an arc of a space curve in rectangular Cartesian coordinates
is equal to
where x, y, z are the current coordinates of a point
If
of the curve.
are parametric equations of the space curve, then the arc length of a section
of it from t = t to t = t
l 2
is
/ a
f 1/7