5th May 2004 Calculus Lecture Dr.Kuenzer
Notes by Omar Shouman
Partial Derivatives
Recall:
Example:
(e.g. is the second coordinate function of )
Example:
The image will be a curve in .
Definition: “Norm”
For , we let
be the norm (or length) of the vector .
In the interpretation of as a point of , its norm is its distance to the origin.
Picture
Definition: “Ball”
Given and, we let:
is a ball with center and radius .
Picture
Definition: (Continuity)
A function is continuous in, if for any , there is such that:
“Small distances remain small”
Definition: (Limit)
Given a function , and , , then:
if for every, there is such that
Remark:
A function f is continuous in if and only if
Definition: (Derivative)
Suppose given a function, and such that there exists an with .
Now, f is differentiable in , if there exists a matrix such that:
with .
Remarks:
1) A function f is differentiable, if f is differentiable in all (allowed) points of D.
2) f is differentiable in if and only if:
are all differentiable in , where:
Now, we consider the matrix in detail:
where (for , )
(Partial derivative of with respect to )
Example: (Find the partial derivatives)
Suppose to be the second coordinate function of a function f.
Finding the partial derivatives:
These partial derivatives form the second row of .
Example: (in which):
Consider
, and
Now, we write the matrix of the partial derivatives in symbols:
=
Substituting in the generalized tangent equation ( the linear approximation):
So the linear approximation is identical to the given function. Therefore, the remainder term is equal to ZERO.
The function is of linear type, therefore the linear approximation is exact, and we do not get a nonzero error term.
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