Unformatted text preview: 3.5. SURFACES AND THEIR TANGENT PLANES 307 and ﬁnding lines tangent to the resulting curves. A more fruitful view,
however, is to think in terms of arbitrary curves in the surface. Suppose
→
− (r, s) is a C 1 function parametrizing the surface S in R3 and
p
→
→
P = − (r0 , s0 ) is a regular point; by restricting the domain of − we can
p
p
assume that we have a coordinate patch for S. Any curve in S can be
represented as
→
− (t) = − (r (t) , s(t))
→
γ
p
or
x = x(r (t) , s(t))
y = y (r (t) , s(t))
z = z (r (t) , s(t))
—that is, we can “pull back” the curve on S to a curve in the parameter
space. If we want the curve to pass through P when t = 0, we need to
require
r (0) = r0
s(0) = s0 .
If r (t) and s(t) are diﬀerentiable, then by the Chain Rule γ (t) is also
diﬀerentiable, and its velocity vector can be found via
→
→
− (t) = − (t)
˙
γ
v
= dx dx dx
,,
dt dt dt where
∂x dr ∂x ds
dx
=
+
dt
∂r dt
∂s dt
dy
∂y dr ∂y ds
=
+
dt
∂r dt
∂s dt
dz
∂z dr ∂z ds
=
+
.
dt
∂r dt
∂s dt
→
We expect that for any such curve, − (0) will be parallel to the tangent
v
plane to S at P . In particular, the two curves obtained by holding one of ...
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 Fall '08
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 Calculus, Derivative, Trigraph, dt dt

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