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

240 Functions of Several Variables [Ch. 6
Whence, when 1, we get
Consequently,
T= '
662
1 2 3
Since for ?=1 we have *=1, y=l, 2=1,
are the equations of the tangent,
are the equations
P = 3/-ay+* V ~ ' -1U-8/+9*
1
""
2 "~
it follows that
3
x\_y l_z 1
3
~~
3 ~~
of the binomial and
*-l y-1 z-1
11 8 9
are the equations of the principal normal.
If a space curve is represented as an intersection of two surfaces
F(JC, y, z) = 0, G(x, y, 2) = 0,
then in place of the vectors -^- and TT- Z we can take the vectors dr{dx, dy, dz}
and d 2 r {d*x, d*y, d z
z}; and one of the variables x, y, z may
independent and we can put its second differential equal to zero.
Example 2. Write the equation of the osculating plane of the circle
1
be considered
* 2 2 + J/ + z 2 = 6, x + y + z^Q (3)
at its point M(l, 1, 2).
Solution. Differentiating the system (3) and considering x an independent
variable, we will have
x dx + y dy + z dz --= 0,
and
dx* + dy 1 + y d*y + dz 2 + z d*z .= 0,
d 2
(/
Putting x=l, y=\> z~2, we get