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

242_Functions of Several Variables_[Ch. 6
2096. Form the equations of the tangent, principal normal,
and binormal at an arbitrary point of the curve
Find the points at which the tangent to this curve is parallel
to the plane x + 3y-\- 2z 10 = 0.
2097. Form equations of the tangent, the osculating plane,
the principal normal and the binormal of the curve
the direction cosines of the binormal
at the point / = 2. Compute
at this point.
2098. Write the equations of the tangent and the normal
plane to the following curves:
z=x z
a) x = R cos 2
/, y = R sin /cos/, z = Rsmt for / = ?-;
b) z=x*+y*, x = y at the point (1,1, 2);
C ) * 2
+ y 2
+ z 2 = 25, x + z = 5 at the point (2, 2/3, 3).
2099 Find the equation of the normal plane
to the curve
if, y = x at the coordinate origin.
2100. Find the equation of the osculating plane to the curve
* = = *, (/ -', 2 = ty2 at the / point = 0.
2101. Find the equations of the osculating plane to the curves:
a) * 2
+y 2
+ 2 2 = 9, x 2
y 2 = 3 at the point (2, 1, 2);
b) * 2 = 4y, x' = 24z at the point (6, 9, 9);
c) JC
2
+ z 2 = a 2
, y 2
fz 2 = 6 2
at any point of the curve (xQJ y z
ot ).
2102. Form the equations of the osculating plane, the principal
normal and the binormal to the curve
y z = x, * 2 = z at the point (1, 1, 1).
2103. Form the equations of the osculating plane, the principal
normal and the binormal to the conical screw-line A; = /COS/,
j/=/sin/, z~bt at the origin. Find the unit vectors of the
tangent, the principal normal, and the binormal at the origin.
Sec. 20. Curvature and Torsion of a Space Curve
1. Curvature. By the curvature of a curve at a point M we mean the
number
/(
* = lim JL,
R AS-+O As f