7Directional derivative
IA Differential Equations
7.2 Stationary points
There is always (at least) one direction in which
df
ds
= 0, namely the direction
parallel to the contour of f. However, local maxima and minima have
df
ds
= 0
for all directions, i.e. ˆs · ∇f = 0 for all ˆs, i.e. ∇f = 0. Then we know that
∂f
∂x
=
∂f
∂y
= 0.
However, apart from maxima and minima, in 3 dimensions, we also have saddle
points:
In general, we define a saddle point to be a point at which
∇f
= 0 but is not a
maximum or minimum.
When we plot out contours of functions, near maxima and minima, the
contours are locally elliptical. Near saddle points, they are locally hyperbolic.
Also, the contour lines cross at and only at saddle points.