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Theorem 5.5
Let
f be a function defined on an open domain
.Suppose that
f has first order partial derivatives at every point of
and that these partial derivatives are continuous at the point (
x_{0},
y_{0}).
Denote
.
Then:

(5.2.1) 
where
and
.
Definition 5.6
A function
f is
differentiable at the point (x_{0},y_{0}) if the partial derivatives
f_{x}(
x_{0},
y_{0}) and
f_{y}(
x_{0},
y_{0}) exist and the equation
2.1
holds for
f at (
x_{0},
y_{0}).
The function f is differentiable on the open domain
if it is
differentiable at every point of
.
Corollary 5.7
If the first order partial derivatives of
f are continuous on the open domain
,
then
f is differentiable on
.
Theorem 5.8
If
f is differentiable at the point (
x_{0},
y_{0}), then
f is continuous
at the point (
x_{0},
y_{0}).
Definition 5.9
The
linearization of a function
f at a point (
x_{0},
y_{0}) is the function
Lf_{(x0,y0)} (x,y)=f(x_{0},y_{0})+ f_{x}(x_{0},y_{0}) (xx_{0})+f_{y}(x_{0},y_{0})(yy_{0})
Example 5.10
Let
and
(
x_{0},
y_{0})=(1,0). Then
and
.
Thus, the linearization of
f at (1,0) is:
We can use this linearization to compute an approximation of
f(1.02, 0.15):
The ``true'' value is
.
Question: Can we have an estimation of the error when using the linearization instead of the actual function?
The positive answer is described in the following result:
Proposition 5.11
Suppose that
f has continuous partial derivatives of order 1 and 2 inan open domain
.
Let
be a rectangle centerd at the point (
x_{0},
y_{0}) and contained in
.
We denote by
M an upper bound for 
f_{xx}, 
f_{xy} and

f_{yy}. Then the error made when replacing
f(
x,
y) by its linearization for
satisfies the following inequality:
Example 5.12
Take
f(
x,
y)=
x^{2}y+
xy^{3}1,
(
x_{0},
y_{0})=(1,2) and
.
We have:
f_{x}=2xy+y^{3},
f_{y}=x^{2}+3xy^{2},
f_{xx}= 2y,
f_{xy}= 2x+y^{2} and
f_{yy}=6xy.
On the rectangle
,
we can take: M=. Then we have the inequality:
As we have
f(1,2)=9, the error percentage is less than
,
i.e. we made an error of at most
.
Next: Partial derivatives of higher
Up: Partial derivatives; the differential
Previous: First partial derivatives.
Noah DanaPicard
20010530