Differentiability for Differentiability for Functions of Two (or Functions of Two (or

more!) Variablesmore!) Variables

Local LinearityLocal Linearity

Recall that when we zoom in on a “sufficiently nice” function of two variables, we see a plane.

Differentiability: A precise Differentiability: A precise definitiondefinition

A function A function ff((xx,,yy) is said to be ) is said to be differentiabledifferentiable at the point at the point ((aa,,bb) provided that there exist real numbers ) provided that there exist real numbers mm and and nn and a function and a function EE((xx,,yy) such that for all ) such that for all xx and and yy

and and

( , ) ( , ) ( ) ( ) ( , )f x y f a b n x a m y b E x y

2 2

( , )0 as (x,y) 0

E x y

x y

What is meant by “sufficiently What is meant by “sufficiently nice”?nice”?

Suppose we zoom in on the function Suppose we zoom in on the function zz==ff((x,yx,y) ) centering our zoom on the point (centering our zoom on the point (aa,,bb) and we see a ) and we see a plane. What can we say about the plane?plane. What can we say about the plane?

The partial derivatives for the plane at the point The partial derivatives for the plane at the point must be the same as the partial derivatives for the must be the same as the partial derivatives for the function---function---In particular, the partial derivatives must all exist!In particular, the partial derivatives must all exist!

The equation for the tangent plane is The equation for the tangent plane is

( , ) ( , ) ( , )( ) ( , )( )f f

L x y f a b a b x a a b y bx y

Partial Derivatives ExistPartial Derivatives Exist

Suppose we have a function

1 if 0 or 0( , )

0 if neither nor is 0

x yf x y

x y

Notice several things:

•The function is not continuous at x=0.•The function is not locally planar at x=0.

•Both partial derivatives exist at x=0.

Directional Derivatives?Directional Derivatives?

It’s not even good enough It’s not even good enough for all of the directional for all of the directional derivatives to exist!derivatives to exist!

Just take a function that is Just take a function that is a bunch of straight lines a bunch of straight lines through the origin with through the origin with random slopes. (One for random slopes. (One for each direction in the each direction in the plane.)plane.)

Directional Derivatives?Directional Derivatives?

If you don’t believe this is If you don’t believe this is a function, just look at it a function, just look at it from “above.”from “above.”

Directional Derivatives?Directional Derivatives?

If you don’t believe this is If you don’t believe this is a function, just look at it a function, just look at it from “above”.from “above”.

There’s one output (There’s one output (zz value) for each input value) for each input (point ((point (xx,,yy)). )).

Differentiability 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).