Consider the surface
S
with equation and let . Then the point lies on S.
What is the rate of change of at () in the direction of an arbitrary unit
vector u
= () ?
Suppose the vertical plane that passes through
P
in the direction of u
intersects S
in a curve C
. If C
has a tangent line at P
then the slope of the tangent line T
to C
at P
is the rate of change of
in the direction of u
, which is called the directional derivative
of
at () in the direction of u
, denoted by
.
On the other hand, if we define a function g
of single variable
t
by
,
then
g'
(0) =
=.
If
is a differentiable function of
and , by Chain Rule, we have
g'
(t
) =,
Put
, we get
.
¡@
So we have the following theorem :
Theorem
If
is a differentiable function of
and , then
has a directional derivative in the direction of any unit vector u
= () and
Note that the theorem is also true if
is a differentiable function of three variables.
In terms of vector functions, the curve
C
obtained by intersection of the vertical plane that passes through P
in the direction of u
with S
can be described by the vector function
F
(t) = (
)
and the tangent vector
T
of C
at P
is given by
F
'(0) = (
).
However,
the directional derivative
in the direction of u
= ( ) at ( ) is the rate of change of at (
) in the direction of u
which is the third component of F
'(0).
Let
, be the curves obtained by intersection of the vertical planes
,
, respectively. Then the tangent vectors of
,
at P
are
= (
) and
= (
), respectively.
The above theorem says that
, which also indicates that
,
,
lies on the same plane ( the tangent plane of S
at P
).
If
is a function of and , then the gradient
of
is the vector function
defined by
So if
is differentiable, then
=
.
u
for any unit vector u
.
If angle between
u
and
(
) is
, then
we have
=
|(
) | | u
|
|
(
) |.
If
is a function of three variables, then the gradient
of
is defined by
and it is also true that if
is differentiable, then f (x
) =(x
)u
for any unit vector u
.
Therefore, we have
Theorem
If
is a function of two or three variables, then the maximum value of the directional derivative f(x
) is
|(x)| and it occurs when u
has the same direction as
(x).
¡@
Let
C
is a level curve of f
, suppose that C
has the vector equation r
(t) = (), that is,
for some constant k
. Ifand f
are all differentiable, by Chain Rule, we get
+
.
¡@
This says that the gradient vector
(
) is perpendicular to the tangent vector r'
() to C
.
The gradient of a function
satisfies the following properties :
1.
points in the direction of maximum increase of the function.
2. || is the rate of increase of
in the direction of maximum increase.
3.
is perpendicular to each level curve of the function
.
4. || is inversely proportional to the spacing between the level curves.
Suppose
S
is a surface with equation
, that is, it is a level surface of a function F
of three variables, let
be a point on S
. Let C
be a curve that lies on the surface S
and passes through point P,
say C
is given by the vector function
r
(t) = (
)
with
r
() = (
) .
Since C
lies on S
, we have
.
Suppose that
and F
are all differentiable, by Chain Rule, we get
+
+
,
that is,
. r
'
(t
) = 0 for allt.
¡@
In particular,
()
.
r'
(
) = 0.
This says that the gradient vector
() is perpendicular to the tangent vector r'
() to any curve C
on S
that passes through P
.
If
(
) is a nonzero vector, then it is a normal vector of the tangent plane to S
at P
. In this case, the equation of the tangent plane is given by
.
The
normal line
to S
at P
, which is the line passing through P and perpendicular to the tangent plane, is given by the symmetric equations
and let
. Then the point
lies on S. What is the rate of change of
at (
) in the direction of an arbitrary unit vector u = (
) ?
in the direction of u , which is called the directional derivative of
at (
) in the direction of u , denoted by
.
,
),
and for u = (
),
.
,
=
.
is a differentiable function of
and
, by Chain Rule, we have
,
, we get
.
is a differentiable function of
and
, then
has a directional derivative in the direction of any unit vector u = (
) and
![D[u]*f(x,y) = f[x]*(x, y)*a+f[y]*(x, y)*b](images/suppl_directionalDerivatives31.gif)
is a differentiable function of three variables.
)
).
in the direction of u = (
) at (
) is the rate of change of
at (
) in the direction of u which is the third component of F '(0).
,
be the curves obtained by intersection of the vertical planes
,
, respectively. Then the tangent vectors of
,
at P are
= (
) and
= (
), respectively.
, which also indicates that
,
,
lies on the same plane ( the tangent plane of S at P ).
![[Maple Plot]](images/suppl_directionalDerivatives54/suppl_directionalDerivatives54_041.jpg)
is a function of
and
, then the gradient of
is the vector function
defined by
, f[y](x,y))](images/suppl_directionalDerivatives61.gif)
is differentiable, then
=
. u for any unit vector u .
) is
, then we have
= |
) | | u |
|
) |.
is a function of three variables, then the gradient of
is defined by
, f[y](x,y,z), f[z](x,y,z))...](images/suppl_directionalDerivatives78.gif)
is differentiable, then
f ( x ) =
u for any unit vector u .
is a function of two or three variables, then the maximum value of the directional derivative
f( x ) is |
(x).
), that is,
for some constant k . If
and f are all differentiable, by Chain Rule, we get
![F[x]](images/suppl_directionalDerivatives90.gif)
+
![F[y]](images/suppl_directionalDerivatives92.gif)
.
) is perpendicular to the tangent vector r ' (
) to C .
![[Maple Plot]](images/suppl_directionalDerivatives98.gif)
![[Maple Plot]](images/suppl_directionalDerivatives101.gif)
satisfies the following properties :
in the direction of maximum increase.
.
, that is, it is a level surface of a function F of three variables, let
be a point on S . Let C be a curve that lies on the surface S and passes through point P, say C is given by the vector function
) with r (
) = (
) .
.
and F are all differentiable, by Chain Rule, we get
![F[x]](images/suppl_directionalDerivatives116.gif)
+
![F[y]](images/suppl_directionalDerivatives118.gif)
+
![F[z]](images/suppl_directionalDerivatives120.gif)
,
) . r ' (
) = 0.
) is perpendicular to the tangent vector r ' (
) to any curve C on S that passes through P .
) is a nonzero vector, then it is a normal vector of the tangent plane to S at P . In this case, the equation of the tangent plane is given by
.
=
.
) to the ellipsoid
,
,
 = 2*z/9](images/suppl_directionalDerivatives142.gif)
,
,
 = -2/3](images/suppl_directionalDerivatives145.gif)
=
![[Maple Plot]](images/suppl_directionalDerivatives11.gif)
![[Maple Plot]](images/suppl_directionalDerivatives136.gif)