A function
is defined on the region
. Let
be the increment of
and
. Then
is called an increment of
and denoted by
.
If is partially differentiable, then
NOTE Note that we can express as
The differential of is denoted by
. Then for
is totally differentiable at
, Let
. Then
, where
can be expressed as
. Thus we write
SOLUTION
1.
2.
3.
Since
, we approximate using these values. Increment of
can be approxiamted with the total differential of
that is
.
SOLUTION
Consider the function
. Let
. Then
. Now let
. Then
. Note that
. Thus
Total Differentiability |
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Use the contrapositive of Theorem4.2, if ![]() ![]() ![]() ![]() |
NOTE Suppose that is totally differentiable at
, then
Check |
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SOLUTION
Then
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By Example4.8,
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Then this limit depends on the value of ![]() ![]() |
Check |
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Alernative Solution By Example4.8, is partially differentiable at
. But by Example4.4,
is not continuous at
. Thus by Theorem4.2,
is not totally diffrentiable at
SOLUTION
and
. Then
is continuous at
. Thus by Theorem4.3,
is totally differentiable at
.
Altenative Solution
and
,
. Then
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A vector orthogonal(perpendiculr) to the plane is called normal vector.
Thus the vector such as
is a normal vector.