Graphics Reference
In-Depth Information
()
=
(
)
+
(
)
Tz
q
,
10
, ,
z
0
, ,
q
z
is the tangent plane at (1,0,2) to the cylinder of radius 1 centered on the z-axis, which
is the surface parameterized by f.
Up to now we have been dealing with the derivative of a function as a linear map
(which is what it is), but the reader probably has also noticed that this is a little
cumbersome and we end up with complicated looking expressions even if we use the
Jacobian matrix. This is not how most people deal with derivatives. To get more of
the look and feel of how most people really work, we use the following consequence
of the chain rule.
4.3.15. Theorem.
Let g
1
,..., g
m
:
R
n
Æ
R
, f :
R
m
Æ
R
be functions that are contin-
uously differentiable at
a
Œ
R
n
and (g
1
(
a
),..., g
m
(
a
)) Œ
R
m
, respectively. Define F :
R
n
Æ
R
by F(
x
) = f(g
1
(
x
),..., g
m
(
x
)). Then
m
Â
()
=
(
()
()
)
( )
DF
a
Df g
a
,...,
g
a
Dg
a
.
(4.5)
i
j
1
m
i
j
j
=
1
Proof.
See [Spiv65].
Equation (4.5), which is really just the chain rule, comes in very handy when
computing derivatives and is usually written informally as
m
∂
∂
f
x
∂
∂
f
y
∂
∂
y
x
Â
1
j
i
=
.
(4.6)
i
j
j
=
Nevertheless, one needs to be aware of the fact that the Df notation of a derivative in
equation (4.5) is more precise and one should always return to that if one has any
problems carrying out a computation. Specifically, although notation of the type
∂f/∂x is the more common notation for the ith partial derivative it can sometimes
be ambiguous whereas the other notation D
i
f(
p
) is
not
. A typical case where ambi-
guities can arise is in the application of the chain rule like in equation (4.6). For
example, if f(u,v) is a function and u = g(x,y) and v = h(x,y), then equation (4.6) turns
into
∂
∂
f
x
∂
∂
f
u
∂
∂
u
x
∂
∂
f
v
∂
∂
v
x
=
+
,
Although this way of writing the chain rule makes it easy to remember the rule, the
formula must be interpreted carefully. For one thing, the f's appearing on the right
hand side of the equation are different from the f on the left hand side. In the first
case, we are considering f to be a function of u and v and, in the second, we are
considering f to be a function of x and y. One should really write
∂
∂
F
x
∂
∂
f
u
∂
∂
u
x
∂
∂
f
v
∂
∂
v
x
=
+
,
