7Directional derivative
IA Differential Equations
7.5 Contours of f(x, y)
Consider
H
in 2 dimensions, and axes in which
H
is diagonal. So
H
=
λ
1
0
0 λ
2
.
Write x − x
0
= (X, Y ).
Then near x
0
,
f
= constant
⇒
x
H
x = constant, i.e.
λ
1
X
2
+
λ
2
Y
2
= constant.
At a maximum or minimum,
λ
1
and
λ
2
have the same sign. So these contours
are locally ellipses. At a saddle point, they have different signs and the contours
are locally hyperbolae.
Example. Find and classify the stationary points of
f
(
x, y
) = 4
x
3
−
12
xy
+
y
2
+ 10y + 6. We have
f
x
= 12x
2
− 12y
f
y
= −12x + 2y + 10
f
xx
= 24x
f
xy
= −12
f
yy
= 2
At stationary points, f
x
= f
y
= 0. So we have
12x
2
− 12y = 0, −12x + 2y + 10 = 0.
The first equation gives
y
=
x
2
. Substituting into the second equation, we obtain
x
= 1
,
5 and
y
= 1
,
25 respectively. So the stationary points are (1
,
1) and (5
,
25)
To classify them, first consider (1
,
1). Our Hessian matrix
H
=
24 −12
−12 2
.
Our signature is
|H
1
|
= 24 and
|H
2
|
=
−
96. Since we have a +
, −
signature, this
an indefinite case and it is a saddle point.
At (5
,
25),
H
=
120 −12
−12 2
So
|H
1
|
= 120 and
|H
2
|
= 240
−
144 = 96.
Since the signature is +, +, it is a minimum.
To draw the contours, we draw what the contours look like near the stationary
points, and then try to join them together, noting that contours cross only at
saddles.
(5,25)
(1,1)
-2 0 2 4 6
-10
0
10
20
30