Transcript Differentiability for Functions of Two (or more!) Variables

Differentiability for Functions
of Two Variables
Local Linearity
Recall that when we zoom in on a “sufficiently
nice” function of two variables, we see a plane.
What is meant by “sufficiently
nice”?
Suppose we zoom in on the function z=f(x,y)
centering our zoom on the point (a,b) and we see a
plane. What can we say about the plane?
• The partial derivatives for the plane at the point
must be the same as the partial derivatives for the
function.
• Therefore, the equation for the tangent plane is
f
f
L( x, y )  f (a, b)  (a, b)( x  a)  (a, b)( y  b)
x
y
In particular. . .
The Partial Derivatives Must Exist
If the partial derivatives don’t exist at the point
(a,b), the function f cannot be locally planar at
(a,b).
Example: (as given in text) A cone
with vertex at the origin cannot be
locally planar there, as it is clear that the
x and y cross sections are not
differentiable there.
Not enough: A Puny Condition
Suppose we have a function
1
f ( x, y)  
0
if x  0 or y  0
if neither x nor y is 0
Whoa! The
existence of the
partial derivatives
doesn’t even
guarantee
continuity at the
point!
Notice several things:
•Both partial derivatives exist at x=0.
•The function is not locally planar at x=0.
•The function is not continuous at x=0.
Directional Derivatives?
It’s not even good enough
for all of the directional
derivatives to exist!
Just take a function that is
a bunch of straight lines
through the origin with
random slopes. (One for
each direction in the
plane.)
Directional Derivatives?
It’s not even good enough
for all of the directional
derivatives to exist!
Locally Planar at the origin?
What do you think?
Directional Derivatives?
If you don’t believe this is
a function, just look at it
from “above”.
There’s one output (z
value) for each input
(point (x,y)).
Differentiability
The function z = f(x,y) is differentiable (locally
planar) at the point (a,b)
if and only if
the partial derivatives of f exist and are
continuous in a small disk centered at (a,b).
Differentiability: A precise definition
A function f(x,y) is said to be differentiable at the point
(a,b) provided that there exist real numbers m and n
and a function E(x,y) such that for all x and y
f ( x, y)  f (a, b)  n( x  a)  m( y  b)  E( x, y)
and
E ( x, y)
x y
2
2
 0 as (x,y)  0
E(x) for One-Variable Functions
E(x) measures the vertical
distance between f (x) and Lp(x)
But E(x)→0 is not enough, even
for functions of one variable!
( x, Lp ( x))
( x, Lp ( x))
E ( x)
E ( x)
( p, f ( p))
( x, f ( x))
What happens to E(x)
as x approaches p?
( p, f ( p))
( x, f ( x))